Theorem caucvgprprlemnjltk 6789

Description: Lemma for caucvgprpr 6810. 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
caucvgprprlemnjltk	⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ (⟨{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,𝑙,𝑝   𝐽,𝑞,𝑢   𝐾,𝑝   𝐾,𝑞   𝑆,𝑞   𝑆,𝑝   𝑢,𝑘   𝑘,𝑙,𝑛   𝑢,𝑛

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝑆(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑞,𝑝,𝑙)   𝐽(𝑘,𝑛)   𝐾(𝑢,𝑘,𝑛,𝑙)

Proof of Theorem caucvgprprlemnjltk

Step	Hyp	Ref	Expression
1		ltsopr 6694	. . 3 ⊢ <P Or P
2		ltrelpr 6603	. . 3 ⊢ <P ⊆ (P × P)
3	1, 2	son2lpi 4721	. 2 ⊢ ¬ (⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<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}⟩)
4		caucvgprprlemnkj.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Q)
5	4	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → 𝑆 ∈ Q)
6		caucvgprprlemnkj.k	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ N)
7	6	ad2antrr 457	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → 𝐾 ∈ N)
8		nnnq 6520	. . . . . . . 8 ⊢ (𝐾 ∈ N → [⟨𝐾, 1𝑜⟩] ~Q ∈ Q)
9		recclnq 6490	. . . . . . . 8 ⊢ ([⟨𝐾, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
10	7, 8, 9	3syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
11		ltaddnq 6505	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
12	5, 10, 11	syl2anc 391	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
13		ltnqpri 6692	. . . . . 6 ⊢ (𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
14	12, 13	syl 14	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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 (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩)
15		simprl 483	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾))
16		caucvgprpr.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:N⟶P)
17		caucvgprpr.cau	. . . . . . . . 9 ⊢ (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}⟩))))
18	16, 17	caucvgprprlemval 6786	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ((𝐹‘𝐽)<P ((𝐹‘𝐾) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
19	18	simprd 107	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
20	19	adantr 261	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
21	1, 2	sotri 4720	. . . . . 6 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P (𝐹‘𝐾) ∧ (𝐹‘𝐾)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
22	15, 20, 21	syl2anc 391	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
23	1, 2	sotri 4720	. . . . 5 ⊢ ((⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩ ∧ ⟨{𝑝 ∣ 𝑝 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))}, {𝑞 ∣ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q 𝑞}⟩<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → ⟨{𝑝 ∣ 𝑝 <Q 𝑆}, {𝑞 ∣ 𝑆 <Q 𝑞}⟩<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
24	14, 22, 23	syl2anc 391	. . . 4 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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 ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
25		simprr 484	. . . 4 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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}⟩)) → ((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}⟩)
26	24, 25	jca 290	. . 3 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ (⟨{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 ((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}⟩))
27	26	ex 108	. 2 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ((⟨{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 ((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}⟩)))
28	3, 27	mtoi 590	1 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ (⟨{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 97   ∈ wcel 1393  {cab 2026  ∀wral 2306  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383  Pcnp 6389   +_P cpp 6391  <_P cltp 6393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-iltp 6568

This theorem is referenced by:  caucvgprprlemnkj  6790