Theorem caucvgprprlemnbj 7717

Description: Lemma for caucvgprpr 7736. 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 7712	. . . . . 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 114	. . . . 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	ffvelcdmd 5669	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐵) ∈ P)
7		recnnpr 7572	. . . . . . . . 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 7561	. . . . . . . 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 411	. . . . . . 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 7572	. . . . . . . 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 7621	. . . . . . 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 411	. . . . . 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 276	. . . . 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 7620	. . . . . 6 ⊢ <P Or P
18		ltrelpr 7529	. . . . . 6 ⊢ <P ⊆ (P × P)
19	17, 18	sotri 5039	. . . . 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 411	. . . 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 7621	. . . . . . . 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 411	. . . . . . 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 276	. . . . . 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 5531	. . . . . . . 8 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
25	24	breq1d 4028	. . . . . . 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 277	. . . . . 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 147	. . . . 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 276	. . . . 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 411	. . . 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 7712	. . . . . 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 112	. . . . 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 7643	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
33	32	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P ∧ 𝑧 ∈ P)) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
34		addcomprg 7602	. . . . . . . . 9 ⊢ ((𝑥 ∈ P ∧ 𝑦 ∈ P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
35	34	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ P ∧ 𝑦 ∈ P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
36	33, 6, 10, 13, 35	caovord2d 6062	. . . . . . 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 147	. . . . . 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 276	. . . . 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 5039	. . . . 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 411	. . . 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 7346	. . . . 5 ⊢ ((𝐵 ∈ N ∧ 𝐽 ∈ N) → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
42	5, 11, 41	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
43	20, 29, 40, 42	mpjao3dan 1318	. . 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	ffvelcdmd 5669	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐽) ∈ P)
45		addclpr 7561	. . . . . 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 411	. . . . 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 4336	. . . . . 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 424	. . . . 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 411	. . . 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 691	. . . 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 134	. . 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 4021	. . . . . . 7 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
54	53	cbvabv 2314	. . . . . 6 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}
55		breq2 4022	. . . . . . 7 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢))
56	55	cbvabv 2314	. . . . . 6 ⊢ {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢}
57	54, 56	opeq12i 3798	. . . . 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 5903	. . . 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 4021	. . . . . 6 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
60	59	cbvabv 2314	. . . . 5 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}
61		breq2 4022	. . . . . 6 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢))
62	61	cbvabv 2314	. . . . 5 ⊢ {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}
63	60, 62	opeq12i 3798	. . . 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 5904	. . 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 4025	. 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 677	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 104   ↔ wb 105   ∨ w3o 979   ∧ w3a 980   = wceq 1364   ∈ wcel 2160  {cab 2175  ∀wral 2468  ⟨cop 3610   class class class wbr 4018   Or wor 4310  ⟶wf 5228  ‘cfv 5232  (class class class)co 5892  1_oc1o 6429  [cec 6552  Ncnpi 7296   <_N clti 7299   ~_Q ceq 7303  *_Qcrq 7308   <_Q cltq 7309  Pcnp 7315   +_P cpp 7317  <_P cltp 7319

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-2o 6437  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7328  df-pli 7329  df-mi 7330  df-lti 7331  df-plpq 7368  df-mpq 7369  df-enq 7371  df-nqqs 7372  df-plqqs 7373  df-mqqs 7374  df-1nqqs 7375  df-rq 7376  df-ltnqqs 7377  df-enq0 7448  df-nq0 7449  df-0nq0 7450  df-plq0 7451  df-mq0 7452  df-inp 7490  df-iplp 7492  df-iltp 7494

This theorem is referenced by:  caucvgprprlemaddq  7732