Theorem distrlem1pr 3921

Description: Lemma for distributive law for positive reals.

Assertion

Ref	Expression
distrlem1pr	⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (A ·P (B +P C)) ⊆ ((A ·P B) +P (A ·P C)))

Proof of Theorem distrlem1pr

Step	Hyp	Ref	Expression
1		df-mp 3883	. . . . . . 7 ⊢ ·P = {⟨⟨y, z⟩, u⟩∣((y ∈ P ∧ z ∈ P) ∧ u = {f∣∃g ∈ y ∃h ∈ z f = (g ·Q h)})}
2		visset 1350	. . . . . . 7 ⊢ w ∈ V
3	1, 2	genpelv 3897	. . . . . 6 ⊢ ((A ∈ P ∧ (B +P C) ∈ P) → (w ∈ (A ·P (B +P C)) ↔ ∃x∃v((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v))))
4		addclpr 3914	. . . . . 6 ⊢ ((B ∈ P ∧ C ∈ P) → (B +P C) ∈ P)
5	3, 4	sylan2 346	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ (A ·P (B +P C)) ↔ ∃x∃v((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v))))
6		df-plp 3882	. . . . . . . . . . 11 ⊢ +P = {⟨⟨x, w⟩, u⟩∣((x ∈ P ∧ w ∈ P) ∧ u = {f∣∃g ∈ x ∃h ∈ w f = (g +Q h)})}
7		visset 1350	. . . . . . . . . . 11 ⊢ v ∈ V
8	6, 7	genpelv 3897	. . . . . . . . . 10 ⊢ ((B ∈ P ∧ C ∈ P) → (v ∈ (B +P C) ↔ ∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z))))
9	8	anbi1d 469	. . . . . . . . 9 ⊢ ((B ∈ P ∧ C ∈ P) → ((v ∈ (B +P C) ∧ w = (x ·Q v)) ↔ (∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
10	9	anbi2d 468	. . . . . . . 8 ⊢ ((B ∈ P ∧ C ∈ P) → ((x ∈ A ∧ (v ∈ (B +P C) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ (∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))))
11		anass 336	. . . . . . . 8 ⊢ (((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v)) ↔ (x ∈ A ∧ (v ∈ (B +P C) ∧ w = (x ·Q v))))
12		19.42vv 968	. . . . . . . . 9 ⊢ (∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ ∃y∃z(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
13		19.41vv 964	. . . . . . . . . 10 ⊢ (∃y∃z(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)) ↔ (∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))
14	13	anbi2i 367	. . . . . . . . 9 ⊢ ((x ∈ A ∧ ∃y∃z(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ (∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
15	12, 14	bitr 151	. . . . . . . 8 ⊢ (∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ (∃y∃z((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
16	10, 11, 15	3bitr4g 428	. . . . . . 7 ⊢ ((B ∈ P ∧ C ∈ P) → (((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v)) ↔ ∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))))
17	16	adantl 305	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v)) ↔ ∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))))
18	17	bi2exdv 938	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (∃x∃v((x ∈ A ∧ v ∈ (B +P C)) ∧ w = (x ·Q v)) ↔ ∃x∃v∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))))
19	5, 18	bitrd 406	. . . 4 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ (A ·P (B +P C)) ↔ ∃x∃v∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)))))
20		exrot4 778	. . . . 5 ⊢ (∃x∃v∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ ∃y∃z∃x∃v(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
21		anass 336	. . . . . . . . . 10 ⊢ ((((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)) ↔ ((y ∈ B ∧ z ∈ C) ∧ (v = (y +Q z) ∧ w = (x ·Q v))))
22	21	biex 733	. . . . . . . . 9 ⊢ (∃v(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)) ↔ ∃v((y ∈ B ∧ z ∈ C) ∧ (v = (y +Q z) ∧ w = (x ·Q v))))
23		19.42v 966	. . . . . . . . 9 ⊢ (∃v((y ∈ B ∧ z ∈ C) ∧ (v = (y +Q z) ∧ w = (x ·Q v))) ↔ ((y ∈ B ∧ z ∈ C) ∧ ∃v(v = (y +Q z) ∧ w = (x ·Q v))))
24		oprex 3018	. . . . . . . . . . 11 ⊢ (y +Q z) ∈ V
25		opreq2 3007	. . . . . . . . . . . 12 ⊢ (v = (y +Q z) → (x ·Q v) = (x ·Q (y +Q z)))
26	25	cleq2d 1112	. . . . . . . . . . 11 ⊢ (v = (y +Q z) → (w = (x ·Q v) ↔ w = (x ·Q (y +Q z))))
27	24, 26	ceqsexv 1371	. . . . . . . . . 10 ⊢ (∃v(v = (y +Q z) ∧ w = (x ·Q v)) ↔ w = (x ·Q (y +Q z)))
28	27	anbi2i 367	. . . . . . . . 9 ⊢ (((y ∈ B ∧ z ∈ C) ∧ ∃v(v = (y +Q z) ∧ w = (x ·Q v))) ↔ ((y ∈ B ∧ z ∈ C) ∧ w = (x ·Q (y +Q z))))
29	22, 23, 28	3bitr 155	. . . . . . . 8 ⊢ (∃v(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v)) ↔ ((y ∈ B ∧ z ∈ C) ∧ w = (x ·Q (y +Q z))))
30	29	anbi2i 367	. . . . . . 7 ⊢ ((x ∈ A ∧ ∃v(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ ((y ∈ B ∧ z ∈ C) ∧ w = (x ·Q (y +Q z)))))
31		19.42v 966	. . . . . . 7 ⊢ (∃v(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ (x ∈ A ∧ ∃v(((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))))
32		anass 336	. . . . . . 7 ⊢ (((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))) ↔ (x ∈ A ∧ ((y ∈ B ∧ z ∈ C) ∧ w = (x ·Q (y +Q z)))))
33	30, 31, 32	3bitr4 158	. . . . . 6 ⊢ (∃v(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))))
34	33	bi3ex 735	. . . . 5 ⊢ (∃y∃z∃x∃v(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ ∃y∃z∃x((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))))
35	20, 34	bitr 151	. . . 4 ⊢ (∃x∃v∃y∃z(x ∈ A ∧ (((y ∈ B ∧ z ∈ C) ∧ v = (y +Q z)) ∧ w = (x ·Q v))) ↔ ∃y∃z∃x((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))))
36	19, 35	syl6bb 414	. . 3 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ (A ·P (B +P C)) ↔ ∃y∃z∃x((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z)))))
37		df-mp 3883	. . . . . . . . . . . 12 ⊢ ·P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {f∣∃g ∈ w ∃h ∈ v f = (g ·Q h)})}
38	37	genpprecl 3898	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → ((x ∈ A ∧ y ∈ B) → (x ·Q y) ∈ (A ·P B)))
39	37	genpprecl 3898	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ C ∈ P) → ((x ∈ A ∧ z ∈ C) → (x ·Q z) ∈ (A ·P C)))
40	38, 39	im2anan9 434	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ y ∈ B) ∧ (x ∈ A ∧ z ∈ C)) → ((x ·Q y) ∈ (A ·P B) ∧ (x ·Q z) ∈ (A ·P C))))
41		df-plp 3882	. . . . . . . . . . . 12 ⊢ +P = {⟨⟨w, v⟩, u⟩∣((w ∈ P ∧ v ∈ P) ∧ u = {f∣∃g ∈ w ∃h ∈ v f = (g +Q h)})}
42	41	genpprecl 3898	. . . . . . . . . . 11 ⊢ (((A ·P B) ∈ P ∧ (A ·P C) ∈ P) → (((x ·Q y) ∈ (A ·P B) ∧ (x ·Q z) ∈ (A ·P C)) → ((x ·Q y) +Q (x ·Q z)) ∈ ((A ·P B) +P (A ·P C))))
43		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ B ∈ P) → (A ·P B) ∈ P)
44		mulclpr 3916	. . . . . . . . . . 11 ⊢ ((A ∈ P ∧ C ∈ P) → (A ·P C) ∈ P)
45	42, 43, 44	syl2an 349	. . . . . . . . . 10 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (((x ·Q y) ∈ (A ·P B) ∧ (x ·Q z) ∈ (A ·P C)) → ((x ·Q y) +Q (x ·Q z)) ∈ ((A ·P B) +P (A ·P C))))
46	40, 45	syld 27	. . . . . . . . 9 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ y ∈ B) ∧ (x ∈ A ∧ z ∈ C)) → ((x ·Q y) +Q (x ·Q z)) ∈ ((A ·P B) +P (A ·P C))))
47		anandi 392	. . . . . . . . 9 ⊢ ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ↔ ((x ∈ A ∧ y ∈ B) ∧ (x ∈ A ∧ z ∈ C)))
48		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
49		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
50	48, 49	distrpq 3861	. . . . . . . . . 10 ⊢ (x ·Q (y +Q z)) = ((x ·Q y) +Q (x ·Q z))
51	50	eleq1i 1152	. . . . . . . . 9 ⊢ ((x ·Q (y +Q z)) ∈ ((A ·P B) +P (A ·P C)) ↔ ((x ·Q y) +Q (x ·Q z)) ∈ ((A ·P B) +P (A ·P C)))
52	46, 47, 51	3imtr4g 426	. . . . . . . 8 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (A ∈ P ∧ C ∈ P)) → ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → (x ·Q (y +Q z)) ∈ ((A ·P B) +P (A ·P C))))
53	52	anandis 394	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → (x ·Q (y +Q z)) ∈ ((A ·P B) +P (A ·P C))))
54		eleq1a 1158	. . . . . . 7 ⊢ ((x ·Q (y +Q z)) ∈ ((A ·P B) +P (A ·P C)) → (w = (x ·Q (y +Q z)) → w ∈ ((A ·P B) +P (A ·P C))))
55	53, 54	syl6 23	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → (w = (x ·Q (y +Q z)) → w ∈ ((A ·P B) +P (A ·P C)))))
56	55	imp3a 279	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))) → w ∈ ((A ·P B) +P (A ·P C))))
57	56	19.23advv 955	. . . 4 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (∃z∃x((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))) → w ∈ ((A ·P B) +P (A ·P C))))
58	57	19.23adv 954	. . 3 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (∃y∃z∃x((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) ∧ w = (x ·Q (y +Q z))) → w ∈ ((A ·P B) +P (A ·P C))))
59	36, 58	sylbid 178	. 2 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (w ∈ (A ·P (B +P C)) → w ∈ ((A ·P B) +P (A ·P C))))
60	59	ssrdv 1509	1 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (A ·P (B +P C)) ⊆ ((A ·P B) +P (A ·P C)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092   ⊆ 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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