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Theorem distrlem4pr 10436
Description: Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
distrlem4pr (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓   𝑥,𝐶,𝑦,𝑧,𝑓

Proof of Theorem distrlem4pr
Dummy variables 𝑤 𝑣 𝑢 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl2 1184 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐵P)
2 simprlr 776 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑦𝐵)
3 elprnq 10401 . . . . 5 ((𝐵P𝑦𝐵) → 𝑦Q)
41, 2, 3syl2anc 584 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑦Q)
5 simp1 1128 . . . . 5 ((𝐴P𝐵P𝐶P) → 𝐴P)
6 simprl 767 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → 𝑓𝐴)
7 elprnq 10401 . . . . 5 ((𝐴P𝑓𝐴) → 𝑓Q)
85, 6, 7syl2an 595 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑓Q)
9 simpl3 1185 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐶P)
10 simprrr 778 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑧𝐶)
11 elprnq 10401 . . . . 5 ((𝐶P𝑧𝐶) → 𝑧Q)
129, 10, 11syl2anc 584 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑧Q)
13 vex 3495 . . . . . 6 𝑥 ∈ V
14 vex 3495 . . . . . 6 𝑓 ∈ V
15 ltmnq 10382 . . . . . 6 (𝑢Q → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
16 vex 3495 . . . . . 6 𝑦 ∈ V
17 mulcomnq 10363 . . . . . 6 (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤)
1813, 14, 15, 16, 17caovord2 7349 . . . . 5 (𝑦Q → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦)))
19 mulclnq 10357 . . . . . 6 ((𝑓Q𝑧Q) → (𝑓 ·Q 𝑧) ∈ Q)
20 ovex 7178 . . . . . . 7 (𝑥 ·Q 𝑦) ∈ V
21 ovex 7178 . . . . . . 7 (𝑓 ·Q 𝑦) ∈ V
22 ltanq 10381 . . . . . . 7 (𝑢Q → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
23 ovex 7178 . . . . . . 7 (𝑓 ·Q 𝑧) ∈ V
24 addcomnq 10361 . . . . . . 7 (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤)
2520, 21, 22, 23, 24caovord2 7349 . . . . . 6 ((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))))
2619, 25syl 17 . . . . 5 ((𝑓Q𝑧Q) → ((𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
2718, 26sylan9bb 510 . . . 4 ((𝑦Q ∧ (𝑓Q𝑧Q)) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
284, 8, 12, 27syl12anc 832 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
29 simpl1 1183 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐴P)
30 addclpr 10428 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
31303adant1 1122 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
3231adantr 481 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝐵 +P 𝐶) ∈ P)
33 mulclpr 10430 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
3429, 32, 33syl2anc 584 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
35 distrnq 10371 . . . . 5 (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))
36 simprrl 777 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑓𝐴)
37 df-plp 10393 . . . . . . . . 9 +P = (𝑢P, 𝑣P ↦ {𝑤 ∣ ∃𝑔𝑢𝑣 𝑤 = (𝑔 +Q )})
38 addclnq 10355 . . . . . . . . 9 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
3937, 38genpprecl 10411 . . . . . . . 8 ((𝐵P𝐶P) → ((𝑦𝐵𝑧𝐶) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)))
4039imp 407 . . . . . . 7 (((𝐵P𝐶P) ∧ (𝑦𝐵𝑧𝐶)) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))
411, 9, 2, 10, 40syl22anc 834 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))
42 df-mp 10394 . . . . . . . 8 ·P = (𝑢P, 𝑣P ↦ {𝑤 ∣ ∃𝑔𝑢𝑣 𝑤 = (𝑔 ·Q )})
43 mulclnq 10357 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
4442, 43genpprecl 10411 . . . . . . 7 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑓𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
4544imp 407 . . . . . 6 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑓𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
4629, 32, 36, 41, 45syl22anc 834 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
4735, 46eqeltrrid 2915 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
48 prcdnq 10403 . . . 4 (((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))))
4934, 47, 48syl2anc 584 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (((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))))
5028, 49sylbid 241 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 <Q 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
51 simpll 763 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → 𝑥𝐴)
52 elprnq 10401 . . . . 5 ((𝐴P𝑥𝐴) → 𝑥Q)
535, 51, 52syl2an 595 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑥Q)
54 vex 3495 . . . . . 6 𝑧 ∈ V
5514, 13, 15, 54, 17caovord2 7349 . . . . 5 (𝑧Q → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧)))
56 mulclnq 10357 . . . . . 6 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) ∈ Q)
57 ltanq 10381 . . . . . 6 ((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))))
5856, 57syl 17 . . . . 5 ((𝑥Q𝑦Q) → ((𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
5955, 58sylan9bbr 511 . . . 4 (((𝑥Q𝑦Q) ∧ 𝑧Q) → (𝑓 <Q 𝑥 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
6053, 4, 12, 59syl21anc 833 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 <Q 𝑥 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
61 distrnq 10371 . . . . 5 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
62 simprll 775 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑥𝐴)
6342, 43genpprecl 10411 . . . . . . 7 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑥𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
6463imp 407 . . . . . 6 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑥𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
6529, 32, 62, 41, 64syl22anc 834 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
6661, 65eqeltrrid 2915 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
67 prcdnq 10403 . . . 4 (((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))))
6834, 66, 67syl2anc 584 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (((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))))
6960, 68sylbid 241 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 <Q 𝑥 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
70 ltsonq 10379 . . . . 5 <Q Or Q
71 sotrieq 5495 . . . . 5 (( <Q Or Q ∧ (𝑥Q𝑓Q)) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
7270, 71mpan 686 . . . 4 ((𝑥Q𝑓Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
7353, 8, 72syl2anc 584 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
74 oveq1 7152 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
7574oveq2d 7161 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
7661, 75syl5eq 2865 . . . . 5 (𝑥 = 𝑓 → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
7776eleq1d 2894 . . . 4 (𝑥 = 𝑓 → ((𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
7865, 77syl5ibcom 246 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
7973, 78sylbird 261 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
8050, 69, 79ecase3d 1026 1 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079  wcel 2105   class class class wbr 5057   Or wor 5466  (class class class)co 7145  Qcnq 10262   +Q cplq 10265   ·Q cmq 10266   <Q cltq 10268  Pcnp 10269   +P cpp 10271   ·P cmp 10272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096  df-er 8278  df-ni 10282  df-pli 10283  df-mi 10284  df-lti 10285  df-plpq 10318  df-mpq 10319  df-ltpq 10320  df-enq 10321  df-nq 10322  df-erq 10323  df-plq 10324  df-mq 10325  df-1nq 10326  df-rq 10327  df-ltnq 10328  df-np 10391  df-plp 10393  df-mp 10394
This theorem is referenced by:  distrlem5pr  10437
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