Theorem caucvgprprlemnbj 7868

Description: Lemma for caucvgprpr 7887. 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 7863	. . . . . 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 5764	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐵) ∈ P)
7		recnnpr 7723	. . . . . . . . 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 7712	. . . . . . . 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 7723	. . . . . . . 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 7772	. . . . . . 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 7771	. . . . . 6 ⊢ <P Or P
18		ltrelpr 7680	. . . . . 6 ⊢ <P ⊆ (P × P)
19	17, 18	sotri 5120	. . . . 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 7772	. . . . . . . 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 5623	. . . . . . . 8 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
25	24	breq1d 4092	. . . . . . 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 7863	. . . . . 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 7794	. . . . . . . . 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 7753	. . . . . . . . 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 6166	. . . . . . 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 5120	. . . . 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 7497	. . . . 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 1341	. . 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 5764	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐽) ∈ P)
45		addclpr 7712	. . . . . 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 4409	. . . . . 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 694	. . . 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 4085	. . . . . . 7 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
54	53	cbvabv 2354	. . . . . 6 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}
55		breq2 4086	. . . . . . 7 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢))
56	55	cbvabv 2354	. . . . . 6 ⊢ {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢}
57	54, 56	opeq12i 3861	. . . . 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 6005	. . . 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 4085	. . . . . 6 ⊢ (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
60	59	cbvabv 2354	. . . . 5 ⊢ {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}
61		breq2 4086	. . . . . 6 ⊢ (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢))
62	61	cbvabv 2354	. . . . 5 ⊢ {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}
63	60, 62	opeq12i 3861	. . . 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 6006	. . 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 4089	. 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 680	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 1001   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ⟨cop 3669   class class class wbr 4082   Or wor 4383  ⟶wf 5310  ‘cfv 5314  (class class class)co 5994  1_oc1o 6545  [cec 6668  Ncnpi 7447   <_N clti 7450   ~_Q ceq 7454  *_Qcrq 7459   <_Q cltq 7460  Pcnp 7466   +_P cpp 7468  <_P cltp 7470

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4377  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-irdg 6506  df-1o 6552  df-2o 6553  df-oadd 6556  df-omul 6557  df-er 6670  df-ec 6672  df-qs 6676  df-ni 7479  df-pli 7480  df-mi 7481  df-lti 7482  df-plpq 7519  df-mpq 7520  df-enq 7522  df-nqqs 7523  df-plqqs 7524  df-mqqs 7525  df-1nqqs 7526  df-rq 7527  df-ltnqqs 7528  df-enq0 7599  df-nq0 7600  df-0nq0 7601  df-plq0 7602  df-mq0 7603  df-inp 7641  df-iplp 7643  df-iltp 7645

This theorem is referenced by:  caucvgprprlemaddq  7883