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Theorem distrlem4prl 7614
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4prl (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(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 distrlem4prl
Dummy variables 𝑤 𝑣 𝑢 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 7431 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
21adantl 277 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
3 simp1 999 . . . . . . 7 ((𝐴P𝐵P𝐶P) → 𝐴P)
4 simpll 527 . . . . . . 7 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → 𝑥 ∈ (1st𝐴))
5 prop 7505 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 elprnql 7511 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
75, 6sylan 283 . . . . . . 7 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
83, 4, 7syl2an 289 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑥Q)
9 simprl 529 . . . . . . 7 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → 𝑓 ∈ (1st𝐴))
10 elprnql 7511 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
115, 10sylan 283 . . . . . . 7 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
123, 9, 11syl2an 289 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑓Q)
13 simpl2 1003 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐵P)
14 simprlr 538 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑦 ∈ (1st𝐵))
15 prop 7505 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
16 elprnql 7511 . . . . . . . 8 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
1715, 16sylan 283 . . . . . . 7 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
1813, 14, 17syl2anc 411 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑦Q)
19 mulcomnqg 7413 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
2019adantl 277 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
212, 8, 12, 18, 20caovord2d 6067 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦)))
22 ltanqg 7430 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
2322adantl 277 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
24 mulclnq 7406 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) ∈ Q)
258, 18, 24syl2anc 411 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q 𝑦) ∈ Q)
26 mulclnq 7406 . . . . . . 7 ((𝑓Q𝑦Q) → (𝑓 ·Q 𝑦) ∈ Q)
2712, 18, 26syl2anc 411 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q 𝑦) ∈ Q)
28 simpl3 1004 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐶P)
29 simprrr 540 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑧 ∈ (1st𝐶))
30 prop 7505 . . . . . . . . 9 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
31 elprnql 7511 . . . . . . . . 9 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (1st𝐶)) → 𝑧Q)
3230, 31sylan 283 . . . . . . . 8 ((𝐶P𝑧 ∈ (1st𝐶)) → 𝑧Q)
3328, 29, 32syl2anc 411 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑧Q)
34 mulclnq 7406 . . . . . . 7 ((𝑓Q𝑧Q) → (𝑓 ·Q 𝑧) ∈ Q)
3512, 33, 34syl2anc 411 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q 𝑧) ∈ Q)
36 addcomnqg 7411 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
3736adantl 277 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
3823, 25, 27, 35, 37caovord2d 6067 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
3921, 38bitrd 188 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
40 simpl1 1002 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐴P)
41 addclpr 7567 . . . . . . . 8 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
42413adant1 1017 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
4342adantr 276 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝐵 +P 𝐶) ∈ P)
44 mulclpr 7602 . . . . . 6 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
4540, 43, 44syl2anc 411 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
46 distrnqg 7417 . . . . . . 7 ((𝑓Q𝑦Q𝑧Q) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
4712, 18, 33, 46syl3anc 1249 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
48 simprrl 539 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑓 ∈ (1st𝐴))
49 df-iplp 7498 . . . . . . . . . 10 +P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 +Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 +Q ))}⟩)
50 addclnq 7405 . . . . . . . . . 10 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5149, 50genpprecll 7544 . . . . . . . . 9 ((𝐵P𝐶P) → ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶))))
5251imp 124 . . . . . . . 8 (((𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))
5313, 28, 14, 29, 52syl22anc 1250 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))
54 df-imp 7499 . . . . . . . . 9 ·P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 ·Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 ·Q ))}⟩)
55 mulclnq 7406 . . . . . . . . 9 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
5654, 55genpprecll 7544 . . . . . . . 8 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑓 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
5756imp 124 . . . . . . 7 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑓 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
5840, 43, 48, 53, 57syl22anc 1250 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
5947, 58eqeltrrd 2267 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
60 prop 7505 . . . . . 6 ((𝐴 ·P (𝐵 +P 𝐶)) ∈ P → ⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P)
61 prcdnql 7514 . . . . . 6 ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(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)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
6260, 61sylan 283 . . . . 5 (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(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)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
6345, 59, 62syl2anc 411 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
6439, 63sylbid 150 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 <Q 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
652, 12, 8, 33, 20caovord2d 6067 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧)))
66 mulclnq 7406 . . . . . . 7 ((𝑥Q𝑧Q) → (𝑥 ·Q 𝑧) ∈ Q)
678, 33, 66syl2anc 411 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q 𝑧) ∈ Q)
68 ltanqg 7430 . . . . . 6 (((𝑓 ·Q 𝑧) ∈ Q ∧ (𝑥 ·Q 𝑧) ∈ Q ∧ (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))))
6935, 67, 25, 68syl3anc 1249 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
7065, 69bitrd 188 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 <Q 𝑥 ↔ ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧))))
71 distrnqg 7417 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
728, 18, 33, 71syl3anc 1249 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
73 simprll 537 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑥 ∈ (1st𝐴))
7454, 55genpprecll 7544 . . . . . . . 8 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑥 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
7574imp 124 . . . . . . 7 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑥 ∈ (1st𝐴) ∧ (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
7640, 43, 73, 53, 75syl22anc 1250 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
7772, 76eqeltrrd 2267 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
78 prcdnql 7514 . . . . . 6 ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘(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)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
7960, 78sylan 283 . . . . 5 (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (1st ‘(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)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8045, 77, 79syl2anc 411 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8170, 80sylbid 150 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 <Q 𝑥 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8264, 81jaod 718 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
83 ltsonq 7428 . . . . 5 <Q Or Q
84 nqtri3or 7426 . . . . 5 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓𝑥 = 𝑓𝑓 <Q 𝑥))
8583, 84sotritrieq 4343 . . . 4 ((𝑥Q𝑓Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
868, 12, 85syl2anc 411 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
87 oveq1 5904 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
8887oveq2d 5913 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
8972, 88sylan9eq 2242 . . . . 5 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
9076adantr 276 . . . . 5 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
9189, 90eqeltrrd 2267 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ 𝑥 = 𝑓) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
9291ex 115 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
9386, 92sylbird 170 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶)))))
94 ltdcnq 7427 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑥 <Q 𝑓)
95 ltdcnq 7427 . . . . . 6 ((𝑓Q𝑥Q) → DECID 𝑓 <Q 𝑥)
9695ancoms 268 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑓 <Q 𝑥)
97 dcor 937 . . . . 5 (DECID 𝑥 <Q 𝑓 → (DECID 𝑓 <Q 𝑥DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
9894, 96, 97sylc 62 . . . 4 ((𝑥Q𝑓Q) → DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥))
998, 12, 98syl2anc 411 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥))
100 df-dc 836 . . 3 (DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥) ↔ ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10199, 100sylib 122 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10282, 93, 101mpjaod 719 1 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835  w3a 980   = wceq 1364  wcel 2160  cop 3610   class class class wbr 4018  cfv 5235  (class class class)co 5897  1st c1st 6164  2nd c2nd 6165  Qcnq 7310   +Q cplq 7312   ·Q cmq 7313   <Q cltq 7315  Pcnp 7321   +P cpp 7323   ·P cmp 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-iplp 7498  df-imp 7499
This theorem is referenced by:  distrlem5prl  7616
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