Theorem distrlem4pr 3924

Description: Lemma for distributive law for positive reals.

Assertion

Ref	Expression
distrlem4pr	⊢ (((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)))

Distinct variable group(s):   x,y,z,f,A   x,B,y,z,f   x,C,y,z,f

Proof of Theorem distrlem4pr

Step	Hyp	Ref	Expression
1		distrlem3pr 3923	. . . . 5 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (f ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (f ∈ Q ∧ (y ∈ Q ∧ z ∈ Q)))
2		visset 1350	. . . . . . . 8 ⊢ x ∈ V
3		visset 1350	. . . . . . . 8 ⊢ f ∈ V
4		visset 1350	. . . . . . . . 9 ⊢ w ∈ V
5		visset 1350	. . . . . . . . 9 ⊢ v ∈ V
6	4, 5	ltmpq 3871	. . . . . . . 8 ⊢ (u ∈ Q → (w <Q v ↔ (u ·Q w) <Q (u ·Q v)))
7		visset 1350	. . . . . . . 8 ⊢ y ∈ V
8	4, 5	mulcompq 3858	. . . . . . . 8 ⊢ (w ·Q v) = (v ·Q w)
9	2, 3, 6, 7, 8	caoprord2 3071	. . . . . . 7 ⊢ (y ∈ Q → (x <Q f ↔ (x ·Q y) <Q (f ·Q y)))
10		mulclpq 3854	. . . . . . . 8 ⊢ ((f ∈ Q ∧ z ∈ Q) → (f ·Q z) ∈ Q)
11		oprex 3018	. . . . . . . . 9 ⊢ (x ·Q y) ∈ V
12		oprex 3018	. . . . . . . . 9 ⊢ (f ·Q y) ∈ V
13	4, 5	ltapq 3870	. . . . . . . . 9 ⊢ (u ∈ Q → (w <Q v ↔ (u +Q w) <Q (u +Q v)))
14		oprex 3018	. . . . . . . . 9 ⊢ (f ·Q z) ∈ V
15	4, 5	addcompq 3856	. . . . . . . . 9 ⊢ (w +Q v) = (v +Q w)
16	11, 12, 13, 14, 15	caoprord2 3071	. . . . . . . 8 ⊢ ((f ·Q z) ∈ Q → ((x ·Q y) <Q (f ·Q y) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
17	10, 16	syl 12	. . . . . . 7 ⊢ ((f ∈ Q ∧ z ∈ Q) → ((x ·Q y) <Q (f ·Q y) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
18	9, 17	sylan9bb 418	. . . . . 6 ⊢ ((y ∈ Q ∧ (f ∈ Q ∧ z ∈ Q)) → (x <Q f ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
19	18	an1s 372	. . . . 5 ⊢ ((f ∈ Q ∧ (y ∈ Q ∧ z ∈ Q)) → (x <Q f ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
20	1, 19	syl 12	. . . 4 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (f ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (x <Q f ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z))))
21		distrlem2pr 3922	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((f ∈ A ∧ (y ∈ B ∧ z ∈ C)) → ((f ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
22		mulclpr 3916	. . . . . . . 8 ⊢ ((A ∈ P ∧ (B +P C) ∈ P) → (A ·P (B +P C)) ∈ P)
23		addclpr 3914	. . . . . . . 8 ⊢ ((B ∈ P ∧ C ∈ P) → (B +P C) ∈ P)
24	22, 23	sylan2 346	. . . . . . 7 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (A ·P (B +P C)) ∈ P)
25		prcdpq 3891	. . . . . . . 8 ⊢ (((A ·P (B +P C)) ∈ P ∧ ((f ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
26	25	exp 291	. . . . . . 7 ⊢ ((A ·P (B +P C)) ∈ P → (((f ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
27	24, 26	syl 12	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((f ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
28	21, 27	syld 27	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((f ∈ A ∧ (y ∈ B ∧ z ∈ C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
29	28	imp 277	. . . 4 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (f ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (((x ·Q y) +Q (f ·Q z)) <Q ((f ·Q y) +Q (f ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
30	20, 29	sylbid 178	. . 3 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (f ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (x <Q f → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
31	30	adantrll 316	. 2 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (x <Q f → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
32		distrlem3pr 3923	. . . . 5 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (x ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (x ∈ Q ∧ (y ∈ Q ∧ z ∈ Q)))
33		visset 1350	. . . . . . . 8 ⊢ z ∈ V
34	3, 2, 6, 33, 8	caoprord2 3071	. . . . . . 7 ⊢ (z ∈ Q → (f <Q x ↔ (f ·Q z) <Q (x ·Q z)))
35		mulclpq 3854	. . . . . . . 8 ⊢ ((x ∈ Q ∧ y ∈ Q) → (x ·Q y) ∈ Q)
36		oprex 3018	. . . . . . . . 9 ⊢ (x ·Q z) ∈ V
37	14, 36	ltapq 3870	. . . . . . . 8 ⊢ ((x ·Q y) ∈ Q → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
38	35, 37	syl 12	. . . . . . 7 ⊢ ((x ∈ Q ∧ y ∈ Q) → ((f ·Q z) <Q (x ·Q z) ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
39	34, 38	sylan9bbr 419	. . . . . 6 ⊢ (((x ∈ Q ∧ y ∈ Q) ∧ z ∈ Q) → (f <Q x ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
40	39	anasss 337	. . . . 5 ⊢ ((x ∈ Q ∧ (y ∈ Q ∧ z ∈ Q)) → (f <Q x ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
41	32, 40	syl 12	. . . 4 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (x ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (f <Q x ↔ ((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z))))
42		distrlem2pr 3922	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → ((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C))))
43		prcdpq 3891	. . . . . . . 8 ⊢ (((A ·P (B +P C)) ∈ P ∧ ((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
44	43	exp 291	. . . . . . 7 ⊢ ((A ·P (B +P C)) ∈ P → (((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
45	24, 44	syl 12	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
46	42, 45	syld 27	. . . . 5 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → ((x ∈ A ∧ (y ∈ B ∧ z ∈ C)) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C)))))
47	46	imp 277	. . . 4 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (x ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (((x ·Q y) +Q (f ·Q z)) <Q ((x ·Q y) +Q (x ·Q z)) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
48	41, 47	sylbid 178	. . 3 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (x ∈ A ∧ (y ∈ B ∧ z ∈ C))) → (f <Q x → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
49	48	adantrlr 317	. 2 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (f <Q x → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
50		elprpq 3889	. . . . . . . 8 ⊢ ((A ∈ P ∧ x ∈ A) → x ∈ Q)
51		elprpq 3889	. . . . . . . 8 ⊢ ((A ∈ P ∧ f ∈ A) → f ∈ Q)
52	50, 51	anim12i 268	. . . . . . 7 ⊢ (((A ∈ P ∧ x ∈ A) ∧ (A ∈ P ∧ f ∈ A)) → (x ∈ Q ∧ f ∈ Q))
53	52	anandis 394	. . . . . 6 ⊢ ((A ∈ P ∧ (x ∈ A ∧ f ∈ A)) → (x ∈ Q ∧ f ∈ Q))
54		ltsopq 3869	. . . . . . 7 ⊢ <Q Or Q
55		sotrieq 2149	. . . . . . 7 ⊢ (( <Q Or Q ∧ (x ∈ Q ∧ f ∈ Q)) → (x = f ↔ ¬ (x <Q f ∨ f <Q x)))
56	54, 55	mpan 518	. . . . . 6 ⊢ ((x ∈ Q ∧ f ∈ Q) → (x = f ↔ ¬ (x <Q f ∨ f <Q x)))
57	53, 56	syl 12	. . . . 5 ⊢ ((A ∈ P ∧ (x ∈ A ∧ f ∈ A)) → (x = f ↔ ¬ (x <Q f ∨ f <Q x)))
58	57	adantlr 310	. . . 4 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ (x ∈ A ∧ f ∈ A)) → (x = f ↔ ¬ (x <Q f ∨ f <Q x)))
59	58	adantrr 312	. . 3 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (x = f ↔ ¬ (x <Q f ∨ f <Q x)))
60		opreq1 3006	. . . . . . 7 ⊢ (x = f → (x ·Q z) = (f ·Q z))
61	60	opreq2d 3013	. . . . . 6 ⊢ (x = f → ((x ·Q y) +Q (x ·Q z)) = ((x ·Q y) +Q (f ·Q z)))
62	61	eleq1d 1155	. . . . 5 ⊢ (x = f → (((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C)) ↔ ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
63		pm3.26 256	. . . . . . 7 ⊢ ((x ∈ A ∧ f ∈ A) → x ∈ A)
64	42, 63	sylani 356	. . . . . 6 ⊢ ((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) → (((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C)) → ((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C))))
65	64	imp 277	. . . . 5 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → ((x ·Q y) +Q (x ·Q z)) ∈ (A ·P (B +P C)))
66	62, 65	syl5bi 183	. . . 4 ⊢ (x = f → (((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))))
67	66	com12 13	. . 3 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (x = f → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
68	59, 67	sylbird 180	. 2 ⊢ (((A ∈ P ∧ (B ∈ P ∧ C ∈ P)) ∧ ((x ∈ A ∧ f ∈ A) ∧ (y ∈ B ∧ z ∈ C))) → (¬ (x <Q f ∨ f <Q x) → ((x ·Q y) +Q (f ·Q z)) ∈ (A ·P (B +P C))))
69	31, 49, 68	ecase3d 560	1 ⊢ (((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)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = weq 797   ∈ wcel 1092   class class class wbr 2054   Or wor 2059  (class class class)co 3001  Qcnq 3773   +_Q cplq 3775   ·_Q cmq 3776   <_Q cltq 3778  Pcnp 3779   +_P cpp 3781   ·_P cmp 3782

This theorem is referenced by:  distrlem5pr 3925

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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