Theorem caucvgprlemnbj 7842

Description: Lemma for caucvgpr 7857. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	⊢ (𝜑 → 𝐹:N⟶Q)
caucvgpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
caucvgprlemnbj.b	⊢ (𝜑 → 𝐵 ∈ N)
caucvgprlemnbj.j	⊢ (𝜑 → 𝐽 ∈ N)

Assertion

Ref	Expression
caucvgprlemnbj	⊢ (𝜑 → ¬ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽))

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

Allowed substitution hints:   𝜑(𝑘,𝑛)

Proof of Theorem caucvgprlemnbj

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

Step	Hyp	Ref	Expression
1		caucvgpr.cau	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))))
2		caucvgprlemnbj.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ N)
3		caucvgprlemnbj.j	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ N)
4		breq1 4085	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → (𝑛 <N 𝑘 ↔ 𝐵 <N 𝑘))
5		fveq2 5623	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → (𝐹‘𝑛) = (𝐹‘𝐵))
6		opeq1 3856	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐵 → ⟨𝑛, 1o⟩ = ⟨𝐵, 1o⟩)
7	6	eceq1d 6706	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐵 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐵, 1o⟩] ~Q )
8	7	fveq2d 5627	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐵, 1o⟩] ~Q ))
9	8	oveq2d 6010	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
10	5, 9	breq12d 4095	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
11	5, 8	oveq12d 6012	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
12	11	breq2d 4094	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
13	10, 12	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → (((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q ))) ↔ ((𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))))
14	4, 13	imbi12d 234	. . . . . . . . 9 ⊢ (𝑛 = 𝐵 → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ (𝐵 <N 𝑘 → ((𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))))
15		breq2 4086	. . . . . . . . . 10 ⊢ (𝑘 = 𝐽 → (𝐵 <N 𝑘 ↔ 𝐵 <N 𝐽))
16		fveq2 5623	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
17	16	oveq1d 6009	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
18	17	breq2d 4094	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
19	16	breq1d 4092	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
20	18, 19	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑘 = 𝐽 → (((𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))) ↔ ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))))
21	15, 20	imbi12d 234	. . . . . . . . 9 ⊢ (𝑘 = 𝐽 → ((𝐵 <N 𝑘 → ((𝐹‘𝐵) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))) ↔ (𝐵 <N 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))))
22	14, 21	rspc2v 2920	. . . . . . . 8 ⊢ ((𝐵 ∈ N ∧ 𝐽 ∈ N) → (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) → (𝐵 <N 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))))
23	2, 3, 22	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) → (𝐵 <N 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))))
24	1, 23	mpd 13	. . . . . 6 ⊢ (𝜑 → (𝐵 <N 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))))
25	24	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → ((𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
26	25	simprd 114	. . . 4 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
27		caucvgpr.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:N⟶Q)
28	27, 2	ffvelcdmd 5764	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵) ∈ Q)
29		nnnq 7597	. . . . . . . 8 ⊢ (𝐵 ∈ N → [⟨𝐵, 1o⟩] ~Q ∈ Q)
30		recclnq 7567	. . . . . . . 8 ⊢ ([⟨𝐵, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q)
31	2, 29, 30	3syl 17	. . . . . . 7 ⊢ (𝜑 → (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q)
32		addclnq 7550	. . . . . . 7 ⊢ (((𝐹‘𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
33	28, 31, 32	syl2anc 411	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
34		nnnq 7597	. . . . . . 7 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
35		recclnq 7567	. . . . . . 7 ⊢ ([⟨𝐽, 1o⟩] ~Q ∈ Q → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
36	3, 34, 35	3syl 17	. . . . . 6 ⊢ (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
37		ltaddnq 7582	. . . . . 6 ⊢ ((((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
38	33, 36, 37	syl2anc 411	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
39	38	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
40		ltsonq 7573	. . . . 5 ⊢ <Q Or Q
41		ltrelnq 7540	. . . . 5 ⊢ <Q ⊆ (Q × Q)
42	40, 41	sotri 5120	. . . 4 ⊢ (((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
43	26, 39, 42	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
44		ltaddnq 7582	. . . . . . 7 ⊢ (((𝐹‘𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → (𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
45	28, 31, 44	syl2anc 411	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
46	45	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
47		fveq2 5623	. . . . . . 7 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
48	47	breq1d 4092	. . . . . 6 ⊢ (𝐵 = 𝐽 → ((𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
49	48	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → ((𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
50	46, 49	mpbid 147	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
51	38	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
52	50, 51, 42	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
53		breq1 4085	. . . . . . . . . 10 ⊢ (𝑛 = 𝐽 → (𝑛 <N 𝑘 ↔ 𝐽 <N 𝑘))
54		fveq2 5623	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (𝐹‘𝑛) = (𝐹‘𝐽))
55		opeq1 3856	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐽 → ⟨𝑛, 1o⟩ = ⟨𝐽, 1o⟩)
56	55	eceq1d 6706	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐽 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
57	56	fveq2d 5627	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
58	57	oveq2d 6010	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
59	54, 58	breq12d 4095	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
60	54, 57	oveq12d 6012	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
61	60	breq2d 4094	. . . . . . . . . . 11 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
62	59, 61	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑛 = 𝐽 → (((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q ))) ↔ ((𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))))
63	53, 62	imbi12d 234	. . . . . . . . 9 ⊢ (𝑛 = 𝐽 → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) ↔ (𝐽 <N 𝑘 → ((𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))))
64		breq2 4086	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (𝐽 <N 𝑘 ↔ 𝐽 <N 𝐵))
65		fveq2 5623	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐹‘𝑘) = (𝐹‘𝐵))
66	65	oveq1d 6009	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
67	66	breq2d 4094	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
68	65	breq1d 4092	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ↔ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
69	67, 68	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (((𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))) ↔ ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))))
70	64, 69	imbi12d 234	. . . . . . . . 9 ⊢ (𝑘 = 𝐵 → ((𝐽 <N 𝑘 → ((𝐹‘𝐽) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))) ↔ (𝐽 <N 𝐵 → ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))))
71	63, 70	rspc2v 2920	. . . . . . . 8 ⊢ ((𝐽 ∈ N ∧ 𝐵 ∈ N) → (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) → (𝐽 <N 𝐵 → ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))))
72	3, 2, 71	syl2anc 411	. . . . . . 7 ⊢ (𝜑 → (∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )))) → (𝐽 <N 𝐵 → ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))))
73	1, 72	mpd 13	. . . . . 6 ⊢ (𝜑 → (𝐽 <N 𝐵 → ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))))
74	73	imp 124	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → ((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹‘𝐵) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
75	74	simpld 112	. . . 4 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → (𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
76		ltanqg 7575	. . . . . . . 8 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
77	76	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
78		addcomnqg 7556	. . . . . . . 8 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
79	78	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
80	77, 28, 33, 36, 79	caovord2d 6166	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
81	45, 80	mpbid 147	. . . . 5 ⊢ (𝜑 → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
82	81	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
83	40, 41	sotri 5120	. . . 4 ⊢ (((𝐹‘𝐽) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
84	75, 82, 83	syl2anc 411	. . 3 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
85		pitri3or 7497	. . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐽 ∈ N) → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
86	2, 3, 85	syl2anc 411	. . 3 ⊢ (𝜑 → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
87	43, 52, 84, 86	mpjao3dan 1341	. 2 ⊢ (𝜑 → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
88	27, 3	ffvelcdmd 5764	. . . 4 ⊢ (𝜑 → (𝐹‘𝐽) ∈ Q)
89		addclnq 7550	. . . . 5 ⊢ ((((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q) → (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q)
90	33, 36, 89	syl2anc 411	. . . 4 ⊢ (𝜑 → (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q)
91		so2nr 4409	. . . . 5 ⊢ (( <Q Or Q ∧ ((𝐹‘𝐽) ∈ Q ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q)) → ¬ ((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)))
92	40, 91	mpan 424	. . . 4 ⊢ (((𝐹‘𝐽) ∈ Q ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q) → ¬ ((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)))
93	88, 90, 92	syl2anc 411	. . 3 ⊢ (𝜑 → ¬ ((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)))
94		imnan 694	. . 3 ⊢ (((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) → ¬ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)) ↔ ¬ ((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)))
95	93, 94	sylibr 134	. 2 ⊢ (𝜑 → ((𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) → ¬ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽)))
96	87, 95	mpd 13	1 ⊢ (𝜑 → ¬ (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (𝐹‘𝐽))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 1001   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀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  Qcnq 7455   +_Q cplq 7457  *_Qcrq 7459   <_Q cltq 7460

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-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

This theorem is referenced by:  caucvgprlemladdrl  7853