ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  caucvgprprlemnkeqj GIF version

Theorem caucvgprprlemnkeqj 7399
Description: Lemma for caucvgprpr 7421. 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 7305 . . . 4 <P Or P
2 ltrelpr 7214 . . . 4 <P ⊆ (P × P)
31, 2son2lpi 4871 . . 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 5488 . . . . . . . 8 (𝜑 → (𝐹𝐽) ∈ P)
76ad2antrr 475 . . . . . . 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 272 . . . . . . . . . . 11 ((𝜑𝐾 = 𝐽) → 𝐽N)
9 nnnq 7131 . . . . . . . . . . 11 (𝐽N → [⟨𝐽, 1o⟩] ~QQ)
108, 9syl 14 . . . . . . . . . 10 ((𝜑𝐾 = 𝐽) → [⟨𝐽, 1o⟩] ~QQ)
11 recclnq 7101 . . . . . . . . . 10 ([⟨𝐽, 1o⟩] ~QQ → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
1210, 11syl 14 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
13 nqprlu 7256 . . . . . . . . 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 272 . . . . . . 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 7306 . . . . . . 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 406 . . . . . 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 502 . . . . . 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 4870 . . . . . 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 406 . . . . 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 272 . . . . . . . . 9 ((𝜑𝐾 = 𝐽) → 𝑆Q)
23 nqprlu 7256 . . . . . . . . 9 (𝑆Q → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
2422, 23syl 14 . . . . . . . 8 ((𝜑𝐾 = 𝐽) → ⟨{𝑝𝑝 <Q 𝑆}, {𝑞𝑆 <Q 𝑞}⟩ ∈ P)
25 ltaddpr 7306 . . . . . . . 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 406 . . . . . . 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 272 . . . . . 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 501 . . . . . . 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 7270 . . . . . . . . . 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 406 . . . . . . . . 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 3885 . . . . . . . 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 272 . . . . . . 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 146 . . . . . 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 4870 . . . . . 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 406 . . . . 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 302 . . . 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 114 . . 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 631 . 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 3652 . . . . . . . . . . 11 (𝐾 = 𝐽 → ⟨𝐾, 1o⟩ = ⟨𝐽, 1o⟩)
4039eceq1d 6395 . . . . . . . . . 10 (𝐾 = 𝐽 → [⟨𝐾, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
4140fveq2d 5357 . . . . . . . . 9 (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
4241oveq2d 5722 . . . . . . . 8 (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
4342breq2d 3887 . . . . . . 7 (𝐾 = 𝐽 → (𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
4443abbidv 2217 . . . . . 6 (𝐾 = 𝐽 → {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))} = {𝑝𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))})
4542breq1d 3885 . . . . . . 7 (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞))
4645abbidv 2217 . . . . . 6 (𝐾 = 𝐽 → {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞})
4744, 46opeq12d 3660 . . . . 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 5353 . . . . 5 (𝐾 = 𝐽 → (𝐹𝐾) = (𝐹𝐽))
4947, 48breq12d 3888 . . . 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 456 . . 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 273 . 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 639 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 103  wb 104   = wceq 1299  wcel 1448  {cab 2086  wral 2375  cop 3477   class class class wbr 3875  wf 5055  cfv 5059  (class class class)co 5706  1oc1o 6236  [cec 6357  Ncnpi 6981   <N clti 6984   ~Q ceq 6988  Qcnq 6989   +Q cplq 6991  *Qcrq 6993   <Q cltq 6994  Pcnp 7000   +P cpp 7002  <P cltp 7004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-2o 6244  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-enq0 7133  df-nq0 7134  df-0nq0 7135  df-plq0 7136  df-mq0 7137  df-inp 7175  df-iplp 7177  df-iltp 7179
This theorem is referenced by:  caucvgprprlemnkj  7401
  Copyright terms: Public domain W3C validator