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Theorem distrlem4prl 7045
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 6862 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
21adantl 271 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
3 simp1 939 . . . . . . 7 ((𝐴P𝐵P𝐶P) → 𝐴P)
4 simpll 496 . . . . . . 7 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → 𝑥 ∈ (1st𝐴))
5 prop 6936 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 elprnql 6942 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (1st𝐴)) → 𝑥Q)
75, 6sylan 277 . . . . . . 7 ((𝐴P𝑥 ∈ (1st𝐴)) → 𝑥Q)
83, 4, 7syl2an 283 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑥Q)
9 simprl 498 . . . . . . 7 (((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶))) → 𝑓 ∈ (1st𝐴))
10 elprnql 6942 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
115, 10sylan 277 . . . . . . 7 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
123, 9, 11syl2an 283 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑓Q)
13 simpl2 943 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐵P)
14 simprlr 505 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑦 ∈ (1st𝐵))
15 prop 6936 . . . . . . . 8 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
16 elprnql 6942 . . . . . . . 8 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (1st𝐵)) → 𝑦Q)
1715, 16sylan 277 . . . . . . 7 ((𝐵P𝑦 ∈ (1st𝐵)) → 𝑦Q)
1813, 14, 17syl2anc 403 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑦Q)
19 mulcomnqg 6844 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
2019adantl 271 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
212, 8, 12, 18, 20caovord2d 5748 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑦) <Q (𝑓 ·Q 𝑦)))
22 ltanqg 6861 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
2322adantl 271 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
24 mulclnq 6837 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) ∈ Q)
258, 18, 24syl2anc 403 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q 𝑦) ∈ Q)
26 mulclnq 6837 . . . . . . 7 ((𝑓Q𝑦Q) → (𝑓 ·Q 𝑦) ∈ Q)
2712, 18, 26syl2anc 403 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q 𝑦) ∈ Q)
28 simpl3 944 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐶P)
29 simprrr 507 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑧 ∈ (1st𝐶))
30 prop 6936 . . . . . . . . 9 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
31 elprnql 6942 . . . . . . . . 9 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (1st𝐶)) → 𝑧Q)
3230, 31sylan 277 . . . . . . . 8 ((𝐶P𝑧 ∈ (1st𝐶)) → 𝑧Q)
3328, 29, 32syl2anc 403 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑧Q)
34 mulclnq 6837 . . . . . . 7 ((𝑓Q𝑧Q) → (𝑓 ·Q 𝑧) ∈ Q)
3512, 33, 34syl2anc 403 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q 𝑧) ∈ Q)
36 addcomnqg 6842 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
3736adantl 271 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
3823, 25, 27, 35, 37caovord2d 5748 . . . . 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 186 . . . 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 942 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝐴P)
41 addclpr 6998 . . . . . . . 8 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
42413adant1 957 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
4342adantr 270 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝐵 +P 𝐶) ∈ P)
44 mulclpr 7033 . . . . . 6 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
4540, 43, 44syl2anc 403 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
46 distrnqg 6848 . . . . . . 7 ((𝑓Q𝑦Q𝑧Q) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
4712, 18, 33, 46syl3anc 1170 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
48 simprrl 506 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑓 ∈ (1st𝐴))
49 df-iplp 6929 . . . . . . . . . 10 +P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 +Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 +Q ))}⟩)
50 addclnq 6836 . . . . . . . . . 10 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
5149, 50genpprecll 6975 . . . . . . . . 9 ((𝐵P𝐶P) → ((𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶)) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶))))
5251imp 122 . . . . . . . 8 (((𝐵P𝐶P) ∧ (𝑦 ∈ (1st𝐵) ∧ 𝑧 ∈ (1st𝐶))) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))
5313, 28, 14, 29, 52syl22anc 1171 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑦 +Q 𝑧) ∈ (1st ‘(𝐵 +P 𝐶)))
54 df-imp 6930 . . . . . . . . 9 ·P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 ·Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 ·Q ))}⟩)
55 mulclnq 6837 . . . . . . . . 9 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
5654, 55genpprecll 6975 . . . . . . . 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 122 . . . . . . 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 1171 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
5947, 58eqeltrrd 2160 . . . . 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 6936 . . . . . 6 ((𝐴 ·P (𝐵 +P 𝐶)) ∈ P → ⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P)
61 prcdnql 6945 . . . . . 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 277 . . . . 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 403 . . . 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 148 . . 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 5748 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑧) <Q (𝑥 ·Q 𝑧)))
66 mulclnq 6837 . . . . . . 7 ((𝑥Q𝑧Q) → (𝑥 ·Q 𝑧) ∈ Q)
678, 33, 66syl2anc 403 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q 𝑧) ∈ Q)
68 ltanqg 6861 . . . . . 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 1170 . . . . 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 186 . . . 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 6848 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
728, 18, 33, 71syl3anc 1170 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
73 simprll 504 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → 𝑥 ∈ (1st𝐴))
7454, 55genpprecll 6975 . . . . . . . 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 122 . . . . . . 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 1171 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (1st ‘(𝐴 ·P (𝐵 +P 𝐶))))
7772, 76eqeltrrd 2160 . . . . 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 6945 . . . . . 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 277 . . . . 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 403 . . . 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 148 . . 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 670 . 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 6859 . . . . 5 <Q Or Q
84 nqtri3or 6857 . . . . 5 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓𝑥 = 𝑓𝑓 <Q 𝑥))
8583, 84sotritrieq 4115 . . . 4 ((𝑥Q𝑓Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
868, 12, 85syl2anc 403 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
87 oveq1 5597 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
8887oveq2d 5606 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
8972, 88sylan9eq 2135 . . . . 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 270 . . . . 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 2160 . . . 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 113 . . 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 168 . 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 6858 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑥 <Q 𝑓)
95 ltdcnq 6858 . . . . . 6 ((𝑓Q𝑥Q) → DECID 𝑓 <Q 𝑥)
9695ancoms 264 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑓 <Q 𝑥)
97 dcor 877 . . . . 5 (DECID 𝑥 <Q 𝑓 → (DECID 𝑓 <Q 𝑥DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
9894, 96, 97sylc 61 . . . 4 ((𝑥Q𝑓Q) → DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥))
998, 12, 98syl2anc 403 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥))
100 df-dc 777 . . 3 (DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥) ↔ ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10199, 100sylib 120 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (1st𝐴) ∧ 𝑦 ∈ (1st𝐵)) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑧 ∈ (1st𝐶)))) → ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10282, 93, 101mpjaod 671 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 102  wb 103  wo 662  DECID wdc 776  w3a 920   = wceq 1285  wcel 1434  cop 3425   class class class wbr 3811  cfv 4968  (class class class)co 5590  1st c1st 5843  2nd c2nd 5844  Qcnq 6741   +Q cplq 6743   ·Q cmq 6744   <Q cltq 6746  Pcnp 6752   +P cpp 6754   ·P cmp 6755
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4079  df-id 4083  df-po 4086  df-iso 4087  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-ov 5593  df-oprab 5594  df-mpt2 5595  df-1st 5845  df-2nd 5846  df-recs 6001  df-irdg 6066  df-1o 6112  df-2o 6113  df-oadd 6116  df-omul 6117  df-er 6221  df-ec 6223  df-qs 6227  df-ni 6765  df-pli 6766  df-mi 6767  df-lti 6768  df-plpq 6805  df-mpq 6806  df-enq 6808  df-nqqs 6809  df-plqqs 6810  df-mqqs 6811  df-1nqqs 6812  df-rq 6813  df-ltnqqs 6814  df-enq0 6885  df-nq0 6886  df-0nq0 6887  df-plq0 6888  df-mq0 6889  df-inp 6927  df-iplp 6929  df-imp 6930
This theorem is referenced by:  distrlem5prl  7047
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