Theorem distrlem5pr 3925

Description: Lemma for distributive law for positive reals.

Assertion

Ref	Expression
distrlem5pr	⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((A ·P B) +P (A ·P C)) ⊆ (A ·P (B +P C)))

Proof of Theorem distrlem5pr

Step	Hyp	Ref	Expression
1		df-plp 3882	. . . . . . 7 ⊢ +P = {⟨⟨x, y⟩, z⟩∣((x ∈ P ∧ y ∈ P) ∧ z = {f∣∃g ∈ x ∃h ∈ y f = (g +Q h)})}
2		visset 1350	. . . . . . 7 ⊢ w ∈ V
3	1, 2	genpelv 3897	. . . . . 6 ⊢ (((A ·P B) ∈ P ∧ (A ·P C) ∈ P) → (w ∈ ((A ·P B) +P (A ·P C)) ↔ ∃v∃u((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u))))
4		mulclpr 3916	. . . . . 6 ⊢ ((A ∈ P ∧ B ∈ P) → (A ·P B) ∈ P)
5		mulclpr 3916	. . . . . 6 ⊢ ((A ∈ P ∧ C ∈ P) → (A ·P C) ∈ P)
6	3, 4, 5	syl2an 349	. . . . 5 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (w ∈ ((A ·P B) +P (A ·P C)) ↔ ∃v∃u((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u))))
7		df-mp 3883	. . . . . . . . . . . 12 ⊢ ·P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {f∣∃g ∈ w ∃h ∈ v f = (g ·Q h)})}
8		visset 1350	. . . . . . . . . . . 12 ⊢ v ∈ V
9	7, 8	genpelv 3897	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → (v ∈ (A ·P B) ↔ ∃x∃y((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y))))
10		df-mp 3883	. . . . . . . . . . . 12 ⊢ ·P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {x∣∃g ∈ w ∃h ∈ v x = (g ·Q h)})}
11		visset 1350	. . . . . . . . . . . 12 ⊢ u ∈ V
12	10, 11	genpelv 3897	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ C ∈ P) → (u ∈ (A ·P C) ↔ ∃f∃z((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))))
13	9, 12	bi2anan9 478	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → ((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ↔ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ∃f∃z((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z)))))
14		ee4anv 982	. . . . . . . . . 10 ⊢ (∃x∃y∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ↔ (∃x∃y((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ∃f∃z((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))))
15	13, 14	syl6bbr 416	. . . . . . . . 9 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → ((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z)))))
16	15	anbi1d 469	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u)) ↔ (∃x∃y∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u))))
17		19.41vv 964	. . . . . . . . . 10 ⊢ (∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ (∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
18	17	bi2ex 734	. . . . . . . . 9 ⊢ (∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃x∃y(∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
19		19.41vv 964	. . . . . . . . 9 ⊢ (∃x∃y(∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ (∃x∃y∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
20	18, 19	bitr 151	. . . . . . . 8 ⊢ (∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ (∃x∃y∃f∃z(((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
21	16, 20	syl6bbr 416	. . . . . . 7 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u)) ↔ ∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u))))
22	21	bi2exdv 938	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (∃v∃u((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u)) ↔ ∃v∃u∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u))))
23		exrot4 778	. . . . . . 7 ⊢ (∃v∃u∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃x∃y∃v∃u∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
24		exrot4 778	. . . . . . . . 9 ⊢ (∃v∃u∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃f∃z∃v∃u((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
25		19.42vv 968	. . . . . . . . . . 11 ⊢ (∃v∃u(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))) ↔ (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ∃v∃u((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))))
26		anass 336	. . . . . . . . . . . . 13 ⊢ (((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))))
27		an4 388	. . . . . . . . . . . . . . . 16 ⊢ (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ↔ ((x ∈ A ∧ y ∈ B) ∧ (f ∈ A ∧ z ∈ C)))
28	27	anbi1i 368	. . . . . . . . . . . . . . 15 ⊢ ((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))) ↔ (((x ∈ A ∧ y ∈ B) ∧ (f ∈ A ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))))
29		an4 388	. . . . . . . . . . . . . . 15 ⊢ ((((x ∈ A ∧ y ∈ B) ∧ (f ∈ A ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))) ↔ (((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))))
30	28, 29	bitr 151	. . . . . . . . . . . . . 14 ⊢ ((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))) ↔ (((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))))
31	30	anbi1i 368	. . . . . . . . . . . . 13 ⊢ (((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ (v = (x ·Q y) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
32	26, 31	bitr3 153	. . . . . . . . . . . 12 ⊢ ((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))) ↔ ((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
33	32	bi2ex 734	. . . . . . . . . . 11 ⊢ (∃v∃u(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))) ↔ ∃v∃u((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)))
34		oprex 3018	. . . . . . . . . . . . 13 ⊢ (x ·Q y) ∈ V
35		oprex 3018	. . . . . . . . . . . . 13 ⊢ (f ·Q z) ∈ V
36		id 9	. . . . . . . . . . . . . 14 ⊢ ((v = (x ·Q y) ∧ u = (f ·Q z)) → (v = (x ·Q y) ∧ u = (f ·Q z)))
37		opreq12 3008	. . . . . . . . . . . . . . 15 ⊢ ((v = (x ·Q y) ∧ u = (f ·Q z)) → (v +Q u) = ((x ·Q y) +Q (f ·Q z)))
38	37	cleq2d 1112	. . . . . . . . . . . . . 14 ⊢ ((v = (x ·Q y) ∧ u = (f ·Q z)) → (w = (v +Q u) ↔ w = ((x ·Q y) +Q (f ·Q z))))
39	36, 38	cgsex2g 1368	. . . . . . . . . . . . 13 ⊢ (((x ·Q y) ∈ V ∧ (f ·Q z) ∈ V) → (∃v∃u((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u)) ↔ w = ((x ·Q y) +Q (f ·Q z))))
40	34, 35, 39	mp2an 520	. . . . . . . . . . . 12 ⊢ (∃v∃u((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u)) ↔ w = ((x ·Q y) +Q (f ·Q z)))
41	40	anbi2i 367	. . . . . . . . . . 11 ⊢ ((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ ∃v∃u((v = (x ·Q y) ∧ u = (f ·Q z)) ∧ w = (v +Q u))) ↔ (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
42	25, 33, 41	3bitr3 156	. . . . . . . . . 10 ⊢ (∃v∃u((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
43	42	bi2ex 734	. . . . . . . . 9 ⊢ (∃f∃z∃v∃u((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
44	24, 43	bitr 151	. . . . . . . 8 ⊢ (∃v∃u∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
45	44	bi2ex 734	. . . . . . 7 ⊢ (∃x∃y∃v∃u∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
46	23, 45	bitr 151	. . . . . 6 ⊢ (∃v∃u∃x∃y∃f∃z((((x ∈ A ∧ y ∈ B) ∧ v = (x ·Q y)) ∧ ((f ∈ A ∧ z ∈ C) ∧ u = (f ·Q z))) ∧ w = (v +Q u)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))))
47	22, 46	syl6bb 414	. . . . 5 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (∃v∃u((v ∈ (A ·P B) ∧ u ∈ (A ·P C)) ∧ w = (v +Q u)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z)))))
48	6, 47	bitrd 406	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (w ∈ ((A ·P B) +P (A ·P C)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z)))))
49	48	anandis 394	. . 3 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ ((A ·P B) +P (A ·P C)) ↔ ∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z)))))
50		eleq1 1149	. . . . . . . . 9 ⊢ (w = ((x ·Q y) +Q (f ·Q z)) → (w ∈ (A ·P (B +P C)) ↔ ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
51		distrlem4pr 3924	. . . . . . . . 9 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))
52	50, 51	syl5bir 184	. . . . . . . 8 ⊢ (w = ((x ·Q y) +Q (f ·Q z)) → (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → w ∈ (A ·P (B +P C))))
53	52	com12 13	. . . . . . 7 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (w = ((x ·Q y) +Q (f ·Q z)) → w ∈ (A ·P (B +P C))))
54	53	exp 291	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) → (w = ((x ·Q y) +Q (f ·Q z)) → w ∈ (A ·P (B +P C)))))
55	54	imp3a 279	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))) → w ∈ (A ·P (B +P C))))
56	55	19.23advv 955	. . . 4 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))) → w ∈ (A ·P (B +P C))))
57	56	19.23advv 955	. . 3 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (∃x∃y∃f∃z(((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) ∧ w = ((x ·Q y) +Q (f ·Q z))) → w ∈ (A ·P (B +P C))))
58	49, 57	sylbid 178	. 2 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ ((A ·P B) +P (A ·P C)) → w ∈ (A ·P (B +P C))))
59	58	ssrdv 1509	1 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((A ·P B) +P (A ·P C)) ⊆ (A ·P (B +P C)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  (class class class)co 3001   +_Q cplq 3775   ·_Q cmq 3776  Pcnp 3779   +_P cpp 3781   ·_P cmp 3782

This theorem is referenced by:  distrpr 3926

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-plp 3882  df-mp 3883

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