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Theorem distrlem4pr 9727
 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 1058 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐵P)
2 simprlr 799 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑦𝐵)
3 elprnq 9692 . . . . 5 ((𝐵P𝑦𝐵) → 𝑦Q)
41, 2, 3syl2anc 691 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑦Q)
5 simp1 1054 . . . . 5 ((𝐴P𝐵P𝐶P) → 𝐴P)
6 simprl 790 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → 𝑓𝐴)
7 elprnq 9692 . . . . 5 ((𝐴P𝑓𝐴) → 𝑓Q)
85, 6, 7syl2an 493 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑓Q)
9 simpl3 1059 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐶P)
10 simprrr 801 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑧𝐶)
11 elprnq 9692 . . . . 5 ((𝐶P𝑧𝐶) → 𝑧Q)
129, 10, 11syl2anc 691 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑧Q)
13 vex 3176 . . . . . 6 𝑥 ∈ V
14 vex 3176 . . . . . 6 𝑓 ∈ V
15 ltmnq 9673 . . . . . 6 (𝑢Q → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
16 vex 3176 . . . . . 6 𝑦 ∈ V
17 mulcomnq 9654 . . . . . 6 (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤)
1813, 14, 15, 16, 17caovord2 6744 . . . . 5 (𝑦Q → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦)))
19 mulclnq 9648 . . . . . 6 ((𝑓Q𝑧Q) → (𝑓 ·Q 𝑧) ∈ Q)
20 ovex 6577 . . . . . . 7 (𝑥 ·Q 𝑦) ∈ V
21 ovex 6577 . . . . . . 7 (𝑓 ·Q 𝑦) ∈ V
22 ltanq 9672 . . . . . . 7 (𝑢Q → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
23 ovex 6577 . . . . . . 7 (𝑓 ·Q 𝑧) ∈ V
24 addcomnq 9652 . . . . . . 7 (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤)
2520, 21, 22, 23, 24caovord2 6744 . . . . . 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 732 . . . 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 1316 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
29 simpl1 1057 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝐴P)
30 addclpr 9719 . . . . . . 7 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
31303adant1 1072 . . . . . 6 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
3231adantr 480 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝐵 +P 𝐶) ∈ P)
33 mulclpr 9721 . . . . 5 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
3429, 32, 33syl2anc 691 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
35 distrnq 9662 . . . . 5 (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))
36 simprrl 800 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑓𝐴)
37 df-plp 9684 . . . . . . . . 9 +P = (𝑢P, 𝑣P ↦ {𝑤 ∣ ∃𝑔𝑢𝑣 𝑤 = (𝑔 +Q )})
38 addclnq 9646 . . . . . . . . 9 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
3937, 38genpprecl 9702 . . . . . . . 8 ((𝐵P𝐶P) → ((𝑦𝐵𝑧𝐶) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)))
4039imp 444 . . . . . . 7 (((𝐵P𝐶P) ∧ (𝑦𝐵𝑧𝐶)) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))
411, 9, 2, 10, 40syl22anc 1319 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶))
42 df-mp 9685 . . . . . . . 8 ·P = (𝑢P, 𝑣P ↦ {𝑤 ∣ ∃𝑔𝑢𝑣 𝑤 = (𝑔 ·Q )})
43 mulclnq 9648 . . . . . . . 8 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
4442, 43genpprecl 9702 . . . . . . 7 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑓𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
4544imp 444 . . . . . 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 1319 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
4735, 46syl5eqelr 2693 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
48 prcdnq 9694 . . . 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 691 . . 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 229 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 <Q 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
51 simpll 786 . . . . 5 (((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶)) → 𝑥𝐴)
52 elprnq 9692 . . . . 5 ((𝐴P𝑥𝐴) → 𝑥Q)
535, 51, 52syl2an 493 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑥Q)
54 vex 3176 . . . . . 6 𝑧 ∈ V
5514, 13, 15, 54, 17caovord2 6744 . . . . 5 (𝑧Q → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧)))
56 mulclnq 9648 . . . . . 6 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) ∈ Q)
57 ltanq 9672 . . . . . 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 733 . . . 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 1317 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 <Q 𝑥 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
61 distrnq 9662 . . . . 5 (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))
62 simprll 798 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → 𝑥𝐴)
6342, 43genpprecl 9702 . . . . . . 7 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑥𝐴 ∧ (𝑦 +Q 𝑧) ∈ (𝐵 +P 𝐶)) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
6463imp 444 . . . . . 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 1319 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
6661, 65syl5eqelr 2693 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶)))
67 prcdnq 9694 . . . 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 691 . . 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 229 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑓 <Q 𝑥 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
70 ltsonq 9670 . . . . 5 <Q Or Q
71 sotrieq 4986 . . . . 5 (( <Q Or Q ∧ (𝑥Q𝑓Q)) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
7270, 71mpan 702 . . . 4 ((𝑥Q𝑓Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
7353, 8, 72syl2anc 691 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
74 oveq1 6556 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
7574oveq2d 6565 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
7661, 75syl5eq 2656 . . . . 5 (𝑥 = 𝑓 → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
7776eleq1d 2672 . . . 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 234 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
7973, 78sylbird 249 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑓𝐴𝑧𝐶))) → (¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (𝐴 ·P (𝐵 +P 𝐶))))
8050, 69, 79ecase3d 981 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 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977   class class class wbr 4583   Or wor 4958  (class class class)co 6549  Qcnq 9553   +Q cplq 9556   ·Q cmq 9557
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