Theorem caucvgprprlemnkeqj 8021

Description: Lemma for caucvgprpr 8043. 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 7927	. . . 4 ⊢ <P Or P
2		ltrelpr 7836	. . . 4 ⊢ <P ⊆ (P × P)
3	1, 2	son2lpi 5164	. . 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 5818	. . . . . . . 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 7753	. . . . . . . . . . 11 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
10	8, 9	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → [⟨𝐽, 1o⟩] ~Q ∈ Q)
11		recclnq 7723	. . . . . . . . . 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 7878	. . . . . . . . 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 7928	. . . . . . 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 533	. . . . . 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 5163	. . . . . 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 7878	. . . . . . . . 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 7928	. . . . . . . 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 531	. . . . . . 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 7892	. . . . . . . . . 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 4124	. . . . . . . 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 5163	. . . . . 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 670	. 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 3888	. . . . . . . . . . 11 ⊢ (𝐾 = 𝐽 → ⟨𝐾, 1o⟩ = ⟨𝐽, 1o⟩)
40	39	eceq1d 6816	. . . . . . . . . 10 ⊢ (𝐾 = 𝐽 → [⟨𝐾, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
41	40	fveq2d 5679	. . . . . . . . 9 ⊢ (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
42	41	oveq2d 6074	. . . . . . . 8 ⊢ (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
43	42	breq2d 4126	. . . . . . 7 ⊢ (𝐾 = 𝐽 → (𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) ↔ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
44	43	abbidv 2354	. . . . . 6 ⊢ (𝐾 = 𝐽 → {𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))} = {𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))})
45	42	breq1d 4124	. . . . . . 7 ⊢ (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞 ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q 𝑞))
46	45	abbidv 2354	. . . . . 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 3896	. . . . 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 5675	. . . . 5 ⊢ (𝐾 = 𝐽 → (𝐹‘𝐾) = (𝐹‘𝐽))
49	47, 48	breq12d 4127	. . . 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 680	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 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ⟨cop 3697   class class class wbr 4114  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  [cec 6778  Ncnpi 7603   <_N clti 7606   ~_Q ceq 7610  Qcnq 7611   +_Q cplq 7613  *_Qcrq 7615   <_Q cltq 7616  Pcnp 7622   +_P cpp 7624  <_P cltp 7626

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-2o 6661  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684  df-enq0 7755  df-nq0 7756  df-0nq0 7757  df-plq0 7758  df-mq0 7759  df-inp 7797  df-iplp 7799  df-iltp 7801

This theorem is referenced by:  caucvgprprlemnkj  8023