Theorem caucvgprprlemnbj 6791

Description: Lemma for caucvgprpr 6810. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-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}⟩))))
caucvgprprlemnbj.b	⊢ (𝜑 → 𝐵 ∈ N)
caucvgprprlemnbj.j	⊢ (𝜑 → 𝐽 ∈ N)

Assertion

Ref	Expression
caucvgprprlemnbj	⊢ (𝜑 → ¬ (((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹‘𝐽))

Distinct variable groups:   𝐵,𝑘,𝑙,𝑛   𝑢,𝐵,𝑘,𝑛   𝑘,𝐹,𝑛   𝑘,𝐽,𝑙,𝑛   𝑢,𝐽

Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑙)

Proof of Theorem caucvgprprlemnbj

Dummy variables 𝑝 𝑞 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:N⟶P)
2		caucvgprpr.cau	. . . . . . 7 ⊢ (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}⟩))))
3	1, 2	caucvgprprlemval 6786	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → ((𝐹‘𝐵)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
4	3	simprd 107	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5		caucvgprprlemnbj.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ N)
6	1, 5	ffvelrnd 5303	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐵) ∈ P)
7		recnnpr 6646	. . . . . . . . 9 ⊢ (𝐵 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
8	5, 7	syl 14	. . . . . . . 8 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
9		addclpr 6635	. . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
10	6, 8, 9	syl2anc 391	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
11		caucvgprprlemnbj.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ N)
12		recnnpr 6646	. . . . . . . 8 ⊢ (𝐽 ∈ N → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
13	11, 12	syl 14	. . . . . . 7 ⊢ (𝜑 → ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
14		ltaddpr 6695	. . . . . . 7 ⊢ ((((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
15	10, 13, 14	syl2anc 391	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
16	15	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
17		ltsopr 6694	. . . . . 6 ⊢ <P Or P
18		ltrelpr 6603	. . . . . 6 ⊢ <P ⊆ (P × P)
19	17, 18	sotri 4720	. . . . 5 ⊢ (((𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
20	4, 16, 19	syl2anc 391	. . . 4 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
21		ltaddpr 6695	. . . . . . . 8 ⊢ (((𝐹‘𝐵) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
22	6, 8, 21	syl2anc 391	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
23	22	adantr 261	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
24		fveq2 5178	. . . . . . . 8 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
25	24	breq1d 3774	. . . . . . 7 ⊢ (𝐵 = 𝐽 → ((𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
26	25	adantl 262	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → ((𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
27	23, 26	mpbid 135	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
28	15	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
29	27, 28, 19	syl2anc 391	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
30	1, 2	caucvgprprlemval 6786	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → ((𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐽) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
31	30	simpld 105	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → (𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
32		ltaprg 6717	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
33	32	adantl 262	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
34		addcomprg 6676	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
35	34	adantl 262	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
36	33, 6, 10, 13, 35	caovord2d 5670	. . . . . . 7 ⊢ (𝜑 → ((𝐹‘𝐵)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
37	22, 36	mpbid 135	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
38	37	adantr 261	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
39	17, 18	sotri 4720	. . . . 5 ⊢ (((𝐹‘𝐽)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
40	31, 38, 39	syl2anc 391	. . . 4 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
41		pitri3or 6420	. . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐽 ∈ N) → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
42	5, 11, 41	syl2anc 391	. . . 4 ⊢ (𝜑 → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
43	20, 29, 40, 42	mpjao3dan 1202	. . 3 ⊢ (𝜑 → (𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
44	1, 11	ffvelrnd 5303	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐽) ∈ P)
45		addclpr 6635	. . . . . 6 ⊢ ((((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
46	10, 13, 45	syl2anc 391	. . . . 5 ⊢ (𝜑 → (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
47		so2nr 4058	. . . . . 6 ⊢ ((<P Or P ∧ ((𝐹‘𝐽) ∈ P ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)) → ¬ ((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
48	17, 47	mpan 400	. . . . 5 ⊢ (((𝐹‘𝐽) ∈ P ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → ¬ ((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
49	44, 46, 48	syl2anc 391	. . . 4 ⊢ (𝜑 → ¬ ((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
50		imnan 624	. . . 4 ⊢ (((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) → ¬ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)) ↔ ¬ ((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
51	49, 50	sylibr 137	. . 3 ⊢ (𝜑 → ((𝐹‘𝐽)<P (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) → ¬ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽)))
52	43, 51	mpd 13	. 2 ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽))
53		breq1 3767	. . . . . . 7 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
54	53	cbvabv 2161	. . . . . 6 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}
55		breq2 3768	. . . . . . 7 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢))
56	55	cbvabv 2161	. . . . . 6 ⊢ {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}
57	54, 56	opeq12i 3554	. . . . 5 ⊢ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩
58	57	oveq2i 5523	. . . 4 ⊢ ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
59		breq1 3767	. . . . . 6 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
60	59	cbvabv 2161	. . . . 5 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}
61		breq2 3768	. . . . . 6 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢))
62	61	cbvabv 2161	. . . . 5 ⊢ {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}
63	60, 62	opeq12i 3554	. . . 4 ⊢ ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩
64	58, 63	oveq12i 5524	. . 3 ⊢ (((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
65	64	breq1i 3771	. 2 ⊢ ((((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹‘𝐽) ↔ (((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹‘𝐽))
66	52, 65	sylnib 601	1 ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ⟨cop 3378   class class class wbr 3764   Or wor 4032  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  *_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-2o 6002  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-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568

This theorem is referenced by:  caucvgprprlemaddq  6806