Theorem caucvgprprlemnkeqj 7691

Description: Lemma for caucvgprpr 7713. Part of disjointness. (Contributed by Jim Kingdon, 12-Feb-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprprlemnkj.k	⊢ (𝜑 → 𝐾 ∈ N)
caucvgprprlemnkj.j	⊢ (𝜑 → 𝐽 ∈ N)
caucvgprprlemnkj.s	⊢ (𝜑 → 𝑆 ∈ Q)

Assertion

Ref	Expression
caucvgprprlemnkeqj	⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <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

Step	Hyp	Ref	Expression
1		ltsopr 7597	. . . 4 ⊢ <P Or P
2		ltrelpr 7506	. . . 4 ⊢ <P ⊆ (P × P)
3	1, 2	son2lpi 5027	. . 3 ⊢ ¬ ((𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽))
4		caucvgprpr.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:N⟶P)
5		caucvgprprlemnkj.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ N)
6	4, 5	ffvelcdmd 5654	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐽) ∈ P)
7	6	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → (𝐹‘𝐽) ∈ P)
8	5	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → 𝐽 ∈ N)
9		nnnq 7423	. . . . . . . . . . 11 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → [⟨𝐽, 1o⟩] ~Q ∈ Q)
11		recclnq 7393	. . . . . . . . . 10 ⊢ ([⟨𝐽, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
12	10, 11	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
13		nqprlu 7548	. . . . . . . . 9 ⊢ ((*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
14	12, 13	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
15	14	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
16		ltaddpr 7598	. . . . . . 7 ⊢ (((𝐹‘𝐽) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐹‘𝐽)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
17	7, 15, 16	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → (𝐹‘𝐽)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
18		simprr 531	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)
19	1, 2	sotri 5026	. . . . . 6 ⊢ (((𝐹‘𝐽)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩) → (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)
20	17, 18, 19	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)
21		caucvgprprlemnkj.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ Q)
22	21	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → 𝑆 ∈ Q)
23		nqprlu 7548	. . . . . . . . 9 ⊢ (𝑆 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∈ P)
24	22, 23	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∈ P)
25		ltaddpr 7598	. . . . . . . 8 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
26	24, 14, 25	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
27	26	adantr 276	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
28		simprl 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽))
29		addnqpr 7562	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
30	22, 12, 29	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
31	30	breq1d 4015	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ↔ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
32	31	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ↔ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
33	28, 32	mpbid 147	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽))
34	1, 2	sotri 5026	. . . . . 6 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽))
35	27, 33, 34	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽))
36	20, 35	jca 306	. . . 4 ⊢ (((𝜑 ∧ 𝐾 = 𝐽) ∧ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)) → ((𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽)))
37	36	ex 115	. . 3 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ((⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩) → ((𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∧ ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽))))
38	3, 37	mtoi 664	. 2 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
39		opeq1 3780	. . . . . . . . . . 11 ⊢ (𝐾 = 𝐽 → ⟨𝐾, 1o⟩ = ⟨𝐽, 1o⟩)
40	39	eceq1d 6573	. . . . . . . . . 10 ⊢ (𝐾 = 𝐽 → [⟨𝐾, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
41	40	fveq2d 5521	. . . . . . . . 9 ⊢ (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
42	41	oveq2d 5893	. . . . . . . 8 ⊢ (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
43	42	breq2d 4017	. . . . . . 7 ⊢ (𝐾 = 𝐽 → (𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
44	43	abbidv 2295	. . . . . 6 ⊢ (𝐾 = 𝐽 → {𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))})
45	42	breq1d 4015	. . . . . . 7 ⊢ (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞))
46	45	abbidv 2295	. . . . . 6 ⊢ (𝐾 = 𝐽 → {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞} = {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞})
47	44, 46	opeq12d 3788	. . . . 5 ⊢ (𝐾 = 𝐽 → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩)
48		fveq2 5517	. . . . 5 ⊢ (𝐾 = 𝐽 → (𝐹‘𝐾) = (𝐹‘𝐽))
49	47, 48	breq12d 4018	. . . 4 ⊢ (𝐾 = 𝐽 → (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ↔ ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽)))
50	49	anbi1d 465	. . 3 ⊢ (𝐾 = 𝐽 → ((⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩) ↔ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)))
51	50	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ((⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩) ↔ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)))
52	38, 51	mtbird 673	1 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ (⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ⟨cop 3597   class class class wbr 4005  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877  1_oc1o 6412  [cec 6535  Ncnpi 7273   <_N clti 7276   ~_Q ceq 7280  Qcnq 7281   +_Q cplq 7283  *_Qcrq 7285   <_Q cltq 7286  Pcnp 7292   +_P cpp 7294  <_P cltp 7296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471

This theorem is referenced by:  caucvgprprlemnkj  7693