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Theorem caucvgprprlemnkeqj 6994
 Description: Lemma for caucvgprpr 7016. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprprlemnkj.k (𝜑𝐾N)
caucvgprprlemnkj.j (𝜑𝐽N)
caucvgprprlemnkj.s (𝜑𝑆Q)
Assertion
Ref Expression
caucvgprprlemnkeqj ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
Distinct variable groups:   𝑘,𝐹,𝑛   𝐽,𝑝,𝑞   𝐾,𝑝,𝑞   𝑆,𝑝,𝑞
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑆(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑞,𝑝,𝑙)   𝐽(𝑢,𝑘,𝑛,𝑙)   𝐾(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprprlemnkeqj
StepHypRef Expression
1 ltsopr 6900 . . . 4 <P Or P
2 ltrelpr 6809 . . . 4 <P ⊆ (P × P)
31, 2son2lpi 4771 . . 3 ¬ ((𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽))
4 caucvgprpr.f . . . . . . . . 9 (𝜑𝐹:NP)
5 caucvgprprlemnkj.j . . . . . . . . 9 (𝜑𝐽N)
64, 5ffvelrnd 5355 . . . . . . . 8 (𝜑 → (𝐹𝐽) ∈ P)
76ad2antrr 472 . . . . . . 7 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽) ∈ P)
85adantr 270 . . . . . . . . . . 11 ((𝜑𝐾 = 𝐽) → 𝐽N)
9 nnnq 6726 . . . . . . . . . . 11 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
108, 9syl 14 . . . . . . . . . 10 ((𝜑𝐾 = 𝐽) → [⟨𝐽, 1𝑜⟩] ~QQ)
11 recclnq 6696 . . . . . . . . . 10 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
1210, 11syl 14 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
13 nqprlu 6851 . . . . . . . . 9 ((*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
1412, 13syl 14 . . . . . . . 8 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
1514adantr 270 . . . . . . 7 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
16 ltaddpr 6901 . . . . . . 7 (((𝐹𝐽) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
177, 15, 16syl2anc 403 . . . . . 6 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
18 simprr 499 . . . . . 6 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
191, 2sotri 4770 . . . . . 6 (((𝐹𝐽)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) → (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
2017, 18, 19syl2anc 403 . . . . 5 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)
21 caucvgprprlemnkj.s . . . . . . . . . 10 (𝜑𝑆Q)
2221adantr 270 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → 𝑆Q)
23 nqprlu 6851 . . . . . . . . 9 (𝑆Q → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
2422, 23syl 14 . . . . . . . 8 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
25 ltaddpr 6901 . . . . . . . 8 ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2624, 14, 25syl2anc 403 . . . . . . 7 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2726adantr 270 . . . . . 6 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
28 simprl 498 . . . . . . 7 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽))
29 addnqpr 6865 . . . . . . . . . 10 ((𝑆Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
3022, 12, 29syl2anc 403 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
3130breq1d 3815 . . . . . . . 8 ((𝜑𝐾 = 𝐽) → (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ↔ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
3231adantr 270 . . . . . . 7 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ↔ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
3328, 32mpbid 145 . . . . . 6 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽))
341, 2sotri 4770 . . . . . 6 ((⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽))
3527, 33, 34syl2anc 403 . . . . 5 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽))
3620, 35jca 300 . . . 4 (((𝜑𝐾 = 𝐽) ∧ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)) → ((𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽)))
3736ex 113 . . 3 ((𝜑𝐾 = 𝐽) → ((⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) → ((𝐹𝐽)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩<P (𝐹𝐽))))
383, 37mtoi 623 . 2 ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
39 opeq1 3590 . . . . . . . . . . 11 (𝐾 = 𝐽 → ⟨𝐾, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
4039eceq1d 6229 . . . . . . . . . 10 (𝐾 = 𝐽 → [⟨𝐾, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
4140fveq2d 5233 . . . . . . . . 9 (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
4241oveq2d 5579 . . . . . . . 8 (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
4342breq2d 3817 . . . . . . 7 (𝐾 = 𝐽 → (𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
4443abbidv 2200 . . . . . 6 (𝐾 = 𝐽 → {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))})
4542breq1d 3815 . . . . . . 7 (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞))
4645abbidv 2200 . . . . . 6 (𝐾 = 𝐽 → {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞})
4744, 46opeq12d 3598 . . . . 5 (𝐾 = 𝐽 → ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
48 fveq2 5229 . . . . 5 (𝐾 = 𝐽 → (𝐹𝐾) = (𝐹𝐽))
4947, 48breq12d 3818 . . . 4 (𝐾 = 𝐽 → (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ↔ ⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽)))
5049anbi1d 453 . . 3 (𝐾 = 𝐽 → ((⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) ↔ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)))
5150adantl 271 . 2 ((𝜑𝐾 = 𝐽) → ((⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩) ↔ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐽) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩)))
5238, 51mtbird 631 1 ((𝜑𝐾 = 𝐽) → ¬ (⟨{𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹𝐾) ∧ ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  {cab 2069  ∀wral 2353  ⟨cop 3419   class class class wbr 3805  ⟶wf 4948  ‘cfv 4952  (class class class)co 5563  1𝑜c1o 6078  [cec 6191  Ncnpi 6576
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