Theorem caucvgprprlemnkltj 7521

Description: Lemma for caucvgprpr 7544. 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
caucvgprprlemnkltj	⊢ ((𝜑 ∧ 𝐾 <N 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   𝐾,𝑙,𝑝   𝑢,𝐾,𝑞   𝑆,𝑝,𝑞   𝑘,𝑙,𝑛   𝑢,𝑘,𝑛

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(𝑘,𝑛)

Proof of Theorem caucvgprprlemnkltj

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltsopr 7428	. . . 4 ⊢ <P Or P
2		ltrelpr 7337	. . . 4 ⊢ <P ⊆ (P × P)
3	1, 2	son2lpi 4943	. . 3 ⊢ ¬ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)
4		simprl 521	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐾))
5		caucvgprpr.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:N⟶P)
6		caucvgprpr.cau	. . . . . . . . . 10 ⊢ (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}⟩))))
7	5, 6	caucvgprprlemval 7520	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ((𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐽)<P ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)))
8	7	simpld 111	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
9	8	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
10	1, 2	sotri 4942	. . . . . . 7 ⊢ (((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐾) ∧ (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)) → (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
11	4, 9, 10	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
12		ltaprg 7451	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
13	12	adantl 275	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P ∧ ℎ ∈ P)) → (𝑓<P 𝑔 ↔ (ℎ +P 𝑓)<P (ℎ +P 𝑔)))
14		caucvgprprlemnkj.s	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ Q)
15	14	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → 𝑆 ∈ Q)
16		nqprlu 7379	. . . . . . . 8 ⊢ (𝑆 ∈ Q → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ ∈ P)
17	15, 16	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩ ∈ P)
18		caucvgprprlemnkj.j	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ N)
19	5, 18	ffvelrnd 5564	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐽) ∈ P)
20	19	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → (𝐹‘𝐽) ∈ P)
21		caucvgprprlemnkj.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ N)
22		recnnpr 7380	. . . . . . . . 9 ⊢ (𝐾 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
23	21, 22	syl 14	. . . . . . . 8 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
24	23	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
25		addcomprg 7410	. . . . . . . 8 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
26	25	adantl 275	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) ∧ (𝑓 ∈ P ∧ 𝑔 ∈ P)) → (𝑓 +P 𝑔) = (𝑔 +P 𝑓))
27	13, 17, 20, 24, 26	caovord2d 5948	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽) ↔ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)))
28	11, 27	mpbird 166	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽))
29		recnnpr 7380	. . . . . . . . 9 ⊢ (𝐽 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
30	18, 29	syl 14	. . . . . . . 8 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
31	30	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
32		ltaddpr 7429	. . . . . . 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}⟩))
33	20, 31, 32	syl2anc 409	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → (𝐹‘𝐽)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
34		simprr 522	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → ((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}⟩)
35	1, 2	sotri 4942	. . . . . 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}⟩)
36	33, 34, 35	syl2anc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)) → (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)
37	28, 36	jca 304	. . . 4 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩))
38	37	ex 114	. . 3 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → (((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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 ∣ 𝑆 <Q 𝑞}⟩<P (𝐹‘𝐽) ∧ (𝐹‘𝐽)<P ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩)))
39	3, 38	mtoi 654	. 2 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ¬ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩))
40	14	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → 𝑆 ∈ Q)
41		nnnq 7254	. . . . . . 7 ⊢ (𝐾 ∈ N → [⟨𝐾, 1o⟩] ~Q ∈ Q)
42		recclnq 7224	. . . . . . 7 ⊢ ([⟨𝐾, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
43	21, 41, 42	3syl 17	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
44	43	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q)
45		addnqpr 7393	. . . . 5 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐾, 1o⟩] ~Q ) ∈ Q) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
46	40, 44, 45	syl2anc 409	. . . 4 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1o⟩] ~Q )) <Q 𝑞}⟩ = (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩))
47	46	breq1d 3947	. . 3 ⊢ ((𝜑 ∧ 𝐾 <N 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 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐾, 1o⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐾)))
48	47	anbi1d 461	. 2 ⊢ ((𝜑 ∧ 𝐾 <N 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 ∣ 𝑆 <Q 𝑞}⟩ +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐾, 1o⟩] ~Q )}, {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}⟩)))
49	39, 48	mtbird 663	1 ⊢ ((𝜑 ∧ 𝐾 <N 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 103   ↔ wb 104   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  {cab 2126  ∀wral 2417  ⟨cop 3535   class class class wbr 3937  ⟶wf 5127  ‘cfv 5131  (class class class)co 5782  1_oc1o 6314  [cec 6435  Ncnpi 7104   <_N clti 7107   ~_Q ceq 7111  Qcnq 7112   +_Q cplq 7114  *_Qcrq 7116   <_Q cltq 7117  Pcnp 7123   +_P cpp 7125  <_P cltp 7127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-eprel 4219  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-1o 6321  df-2o 6322  df-oadd 6325  df-omul 6326  df-er 6437  df-ec 6439  df-qs 6443  df-ni 7136  df-pli 7137  df-mi 7138  df-lti 7139  df-plpq 7176  df-mpq 7177  df-enq 7179  df-nqqs 7180  df-plqqs 7181  df-mqqs 7182  df-1nqqs 7183  df-rq 7184  df-ltnqqs 7185  df-enq0 7256  df-nq0 7257  df-0nq0 7258  df-plq0 7259  df-mq0 7260  df-inp 7298  df-iplp 7300  df-iltp 7302

This theorem is referenced by:  caucvgprprlemnkj  7524