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Theorem caucvgprprlemnkeqj 6845
Description: Lemma for caucvgprpr 6867. 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 6751 . . . 4 <P Or P
2 ltrelpr 6660 . . . 4 <P ⊆ (P × P)
31, 2son2lpi 4748 . . 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 5330 . . . . . . . 8 (𝜑 → (𝐹𝐽) ∈ P)
76ad2antrr 465 . . . . . . 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 265 . . . . . . . . . . 11 ((𝜑𝐾 = 𝐽) → 𝐽N)
9 nnnq 6577 . . . . . . . . . . 11 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
108, 9syl 14 . . . . . . . . . 10 ((𝜑𝐾 = 𝐽) → [⟨𝐽, 1𝑜⟩] ~QQ)
11 recclnq 6547 . . . . . . . . . 10 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
1210, 11syl 14 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
13 nqprlu 6702 . . . . . . . . 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 265 . . . . . . 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 6752 . . . . . . 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 397 . . . . . 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 492 . . . . . 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 4747 . . . . . 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 397 . . . . 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 265 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → 𝑆Q)
23 nqprlu 6702 . . . . . . . . 9 (𝑆Q → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
2422, 23syl 14 . . . . . . . 8 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
25 ltaddpr 6752 . . . . . . . 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 397 . . . . . . 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 265 . . . . . 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 491 . . . . . . 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 6716 . . . . . . . . . 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 397 . . . . . . . . 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 3801 . . . . . . . 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 265 . . . . . . 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 139 . . . . . 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 4747 . . . . . 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 397 . . . . 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 294 . . . 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 112 . . 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 600 . 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 3576 . . . . . . . . . . 11 (𝐾 = 𝐽 → ⟨𝐾, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
4039eceq1d 6172 . . . . . . . . . 10 (𝐾 = 𝐽 → [⟨𝐾, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
4140fveq2d 5209 . . . . . . . . 9 (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
4241oveq2d 5555 . . . . . . . 8 (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
4342breq2d 3803 . . . . . . 7 (𝐾 = 𝐽 → (𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
4443abbidv 2171 . . . . . 6 (𝐾 = 𝐽 → {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))} = {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))})
4542breq1d 3801 . . . . . . 7 (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞 ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞))
4645abbidv 2171 . . . . . 6 (𝐾 = 𝐽 → {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑞})
4744, 46opeq12d 3584 . . . . 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 5205 . . . . 5 (𝐾 = 𝐽 → (𝐹𝐾) = (𝐹𝐽))
4947, 48breq12d 3804 . . . 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 446 . . 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 266 . 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 608 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 101  wb 102   = wceq 1259  wcel 1409  {cab 2042  wral 2323  cop 3405   class class class wbr 3791  wf 4925  cfv 4929  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440  Pcnp 6446   +P cpp 6448  <P cltp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-iplp 6623  df-iltp 6625
This theorem is referenced by:  caucvgprprlemnkj  6847
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