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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
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