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Theorem distrlem4pru 7407
Description: Lemma for distributive law for positive reals. (Contributed by Jim Kingdon, 12-Dec-2019.)
Assertion
Ref Expression
distrlem4pru (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(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 distrlem4pru
Dummy variables 𝑤 𝑣 𝑢 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 7223 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
21adantl 275 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 ·Q 𝑤) <Q (𝑢 ·Q 𝑣)))
3 simp1 981 . . . . . . 7 ((𝐴P𝐵P𝐶P) → 𝐴P)
4 simpll 518 . . . . . . 7 (((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶))) → 𝑥 ∈ (2nd𝐴))
5 prop 7297 . . . . . . . 8 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
6 elprnqu 7304 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
75, 6sylan 281 . . . . . . 7 ((𝐴P𝑥 ∈ (2nd𝐴)) → 𝑥Q)
83, 4, 7syl2an 287 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑥Q)
9 simprl 520 . . . . . . 7 (((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶))) → 𝑓 ∈ (2nd𝐴))
10 elprnqu 7304 . . . . . . . 8 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
115, 10sylan 281 . . . . . . 7 ((𝐴P𝑓 ∈ (2nd𝐴)) → 𝑓Q)
123, 9, 11syl2an 287 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑓Q)
13 simpl3 986 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝐶P)
14 simprrr 529 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑧 ∈ (2nd𝐶))
15 prop 7297 . . . . . . . 8 (𝐶P → ⟨(1st𝐶), (2nd𝐶)⟩ ∈ P)
16 elprnqu 7304 . . . . . . . 8 ((⟨(1st𝐶), (2nd𝐶)⟩ ∈ P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
1715, 16sylan 281 . . . . . . 7 ((𝐶P𝑧 ∈ (2nd𝐶)) → 𝑧Q)
1813, 14, 17syl2anc 408 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑧Q)
19 mulcomnqg 7205 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
2019adantl 275 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 ·Q 𝑣) = (𝑣 ·Q 𝑤))
212, 8, 12, 18, 20caovord2d 5940 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 <Q 𝑓 ↔ (𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧)))
22 mulclnq 7198 . . . . . . 7 ((𝑥Q𝑧Q) → (𝑥 ·Q 𝑧) ∈ Q)
238, 18, 22syl2anc 408 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 ·Q 𝑧) ∈ Q)
24 mulclnq 7198 . . . . . . 7 ((𝑓Q𝑧Q) → (𝑓 ·Q 𝑧) ∈ Q)
2512, 18, 24syl2anc 408 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 ·Q 𝑧) ∈ Q)
26 simpl2 985 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝐵P)
27 simprlr 527 . . . . . . . 8 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑦 ∈ (2nd𝐵))
28 prop 7297 . . . . . . . . 9 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
29 elprnqu 7304 . . . . . . . . 9 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
3028, 29sylan 281 . . . . . . . 8 ((𝐵P𝑦 ∈ (2nd𝐵)) → 𝑦Q)
3126, 27, 30syl2anc 408 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑦Q)
32 mulclnq 7198 . . . . . . 7 ((𝑥Q𝑦Q) → (𝑥 ·Q 𝑦) ∈ Q)
338, 31, 32syl2anc 408 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 ·Q 𝑦) ∈ Q)
34 ltanqg 7222 . . . . . 6 (((𝑥 ·Q 𝑧) ∈ Q ∧ (𝑓 ·Q 𝑧) ∈ Q ∧ (𝑥 ·Q 𝑦) ∈ Q) → ((𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
3523, 25, 33, 34syl3anc 1216 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑧) <Q (𝑓 ·Q 𝑧) ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
3621, 35bitrd 187 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 <Q 𝑓 ↔ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
37 simpl1 984 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝐴P)
38 addclpr 7359 . . . . . . . 8 ((𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
39383adant1 999 . . . . . . 7 ((𝐴P𝐵P𝐶P) → (𝐵 +P 𝐶) ∈ P)
4039adantr 274 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝐵 +P 𝐶) ∈ P)
41 mulclpr 7394 . . . . . 6 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
4237, 40, 41syl2anc 408 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝐴 ·P (𝐵 +P 𝐶)) ∈ P)
43 distrnqg 7209 . . . . . . 7 ((𝑥Q𝑦Q𝑧Q) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
448, 31, 18, 43syl3anc 1216 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)))
45 simprll 526 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑥 ∈ (2nd𝐴))
46 df-iplp 7290 . . . . . . . . . 10 +P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 +Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 +Q ))}⟩)
47 addclnq 7197 . . . . . . . . . 10 ((𝑔QQ) → (𝑔 +Q ) ∈ Q)
4846, 47genppreclu 7337 . . . . . . . . 9 ((𝐵P𝐶P) → ((𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶)) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))))
4948imp 123 . . . . . . . 8 (((𝐵P𝐶P) ∧ (𝑦 ∈ (2nd𝐵) ∧ 𝑧 ∈ (2nd𝐶))) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))
5026, 13, 27, 14, 49syl22anc 1217 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))
51 df-imp 7291 . . . . . . . . 9 ·P = (𝑢P, 𝑣P ↦ ⟨{𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (1st𝑢) ∧ ∈ (1st𝑣) ∧ 𝑤 = (𝑔 ·Q ))}, {𝑤Q ∣ ∃𝑔QQ (𝑔 ∈ (2nd𝑢) ∧ ∈ (2nd𝑣) ∧ 𝑤 = (𝑔 ·Q ))}⟩)
52 mulclnq 7198 . . . . . . . . 9 ((𝑔QQ) → (𝑔 ·Q ) ∈ Q)
5351, 52genppreclu 7337 . . . . . . . 8 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑥 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
5453imp 123 . . . . . . 7 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑥 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
5537, 40, 45, 50, 54syl22anc 1217 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
5644, 55eqeltrrd 2217 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
57 prop 7297 . . . . . 6 ((𝐴 ·P (𝐵 +P 𝐶)) ∈ P → ⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P)
58 prcunqu 7307 . . . . . 6 ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
5957, 58sylan 281 . . . . 5 (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
6042, 56, 59syl2anc 408 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
6136, 60sylbid 149 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 <Q 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
622, 12, 8, 31, 20caovord2d 5940 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 <Q 𝑥 ↔ (𝑓 ·Q 𝑦) <Q (𝑥 ·Q 𝑦)))
63 ltanqg 7222 . . . . . . 7 ((𝑤Q𝑣Q𝑢Q) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
6463adantl 275 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ (𝑤Q𝑣Q𝑢Q)) → (𝑤 <Q 𝑣 ↔ (𝑢 +Q 𝑤) <Q (𝑢 +Q 𝑣)))
65 mulclnq 7198 . . . . . . 7 ((𝑓Q𝑦Q) → (𝑓 ·Q 𝑦) ∈ Q)
6612, 31, 65syl2anc 408 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 ·Q 𝑦) ∈ Q)
67 addcomnqg 7203 . . . . . . 7 ((𝑤Q𝑣Q) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
6867adantl 275 . . . . . 6 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ (𝑤Q𝑣Q)) → (𝑤 +Q 𝑣) = (𝑣 +Q 𝑤))
6964, 66, 33, 25, 68caovord2d 5940 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑓 ·Q 𝑦) <Q (𝑥 ·Q 𝑦) ↔ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
7062, 69bitrd 187 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 <Q 𝑥 ↔ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧))))
71 distrnqg 7209 . . . . . . 7 ((𝑓Q𝑦Q𝑧Q) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
7212, 31, 18, 71syl3anc 1216 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) = ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
73 simprrl 528 . . . . . . 7 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → 𝑓 ∈ (2nd𝐴))
7451, 52genppreclu 7337 . . . . . . . 8 ((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) → ((𝑓 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
7574imp 123 . . . . . . 7 (((𝐴P ∧ (𝐵 +P 𝐶) ∈ P) ∧ (𝑓 ∈ (2nd𝐴) ∧ (𝑦 +Q 𝑧) ∈ (2nd ‘(𝐵 +P 𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
7637, 40, 73, 50, 75syl22anc 1217 . . . . . 6 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
7772, 76eqeltrrd 2217 . . . . 5 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
78 prcunqu 7307 . . . . . 6 ((⟨(1st ‘(𝐴 ·P (𝐵 +P 𝐶))), (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))⟩ ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
7957, 78sylan 281 . . . . 5 (((𝐴 ·P (𝐵 +P 𝐶)) ∈ P ∧ ((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8042, 77, 79syl2anc 408 . . . 4 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (((𝑓 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) <Q ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8170, 80sylbid 149 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑓 <Q 𝑥 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
8261, 81jaod 706 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
83 ltsonq 7220 . . . . 5 <Q Or Q
84 nqtri3or 7218 . . . . 5 ((𝑥Q𝑓Q) → (𝑥 <Q 𝑓𝑥 = 𝑓𝑓 <Q 𝑥))
8583, 84sotritrieq 4247 . . . 4 ((𝑥Q𝑓Q) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
868, 12, 85syl2anc 408 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 = 𝑓 ↔ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
87 oveq1 5781 . . . . . . 7 (𝑥 = 𝑓 → (𝑥 ·Q 𝑧) = (𝑓 ·Q 𝑧))
8887oveq2d 5790 . . . . . 6 (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑥 ·Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
8944, 88sylan9eq 2192 . . . . 5 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) = ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)))
9055adantr 274 . . . . 5 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ 𝑥 = 𝑓) → (𝑥 ·Q (𝑦 +Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
9189, 90eqeltrrd 2217 . . . 4 ((((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) ∧ 𝑥 = 𝑓) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
9291ex 114 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (𝑥 = 𝑓 → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
9386, 92sylbird 169 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → (¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶)))))
94 ltdcnq 7219 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑥 <Q 𝑓)
95 ltdcnq 7219 . . . . . 6 ((𝑓Q𝑥Q) → DECID 𝑓 <Q 𝑥)
9695ancoms 266 . . . . 5 ((𝑥Q𝑓Q) → DECID 𝑓 <Q 𝑥)
97 dcor 919 . . . . 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 408 . . 3 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥))
100 df-dc 820 . . 3 (DECID (𝑥 <Q 𝑓𝑓 <Q 𝑥) ↔ ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10199, 100sylib 121 . 2 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 <Q 𝑓𝑓 <Q 𝑥) ∨ ¬ (𝑥 <Q 𝑓𝑓 <Q 𝑥)))
10282, 93, 101mpjaod 707 1 (((𝐴P𝐵P𝐶P) ∧ ((𝑥 ∈ (2nd𝐴) ∧ 𝑦 ∈ (2nd𝐵)) ∧ (𝑓 ∈ (2nd𝐴) ∧ 𝑧 ∈ (2nd𝐶)))) → ((𝑥 ·Q 𝑦) +Q (𝑓 ·Q 𝑧)) ∈ (2nd ‘(𝐴 ·P (𝐵 +P 𝐶))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  DECID wdc 819  w3a 962   = wceq 1331  wcel 1480  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  Qcnq 7102   +Q cplq 7104   ·Q cmq 7105   <Q cltq 7107  Pcnp 7113   +P cpp 7115   ·P cmp 7116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iplp 7290  df-imp 7291
This theorem is referenced by:  distrlem5pru  7409
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