Theorem caucvgprlemnbj 7998

Description: Lemma for caucvgpr 8013. 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 4117	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → (𝑛 <N 𝑘 ↔ 𝐵 <N 𝑘))
5		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → (𝐹‘𝑛) = (𝐹‘𝐵))
6		opeq1 3888	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐵 → ⟨𝑛, 1o⟩ = ⟨𝐵, 1o⟩)
7	6	eceq1d 6816	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐵 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐵, 1o⟩] ~Q )
8	7	fveq2d 5679	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐵, 1o⟩] ~Q ))
9	8	oveq2d 6074	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
10	5, 9	breq12d 4127	. . . . . . . . . . 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 6076	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
12	11	breq2d 4126	. . . . . . . . . . 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 4118	. . . . . . . . . 10 ⊢ (𝑘 = 𝐽 → (𝐵 <N 𝑘 ↔ 𝐵 <N 𝐽))
16		fveq2 5675	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
17	16	oveq1d 6073	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
18	17	breq2d 4126	. . . . . . . . . . 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 4124	. . . . . . . . . . 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 2937	. . . . . . . 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 5818	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵) ∈ Q)
29		nnnq 7753	. . . . . . . 8 ⊢ (𝐵 ∈ N → [⟨𝐵, 1o⟩] ~Q ∈ Q)
30		recclnq 7723	. . . . . . . 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 7706	. . . . . . 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 7753	. . . . . . 7 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
35		recclnq 7723	. . . . . . 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 7738	. . . . . 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 7729	. . . . 5 ⊢ <Q Or Q
41		ltrelnq 7696	. . . . 5 ⊢ <Q ⊆ (Q × Q)
42	40, 41	sotri 5163	. . . 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 7738	. . . . . . 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 5675	. . . . . . 7 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
48	47	breq1d 4124	. . . . . 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 4117	. . . . . . . . . 10 ⊢ (𝑛 = 𝐽 → (𝑛 <N 𝑘 ↔ 𝐽 <N 𝑘))
54		fveq2 5675	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (𝐹‘𝑛) = (𝐹‘𝐽))
55		opeq1 3888	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐽 → ⟨𝑛, 1o⟩ = ⟨𝐽, 1o⟩)
56	55	eceq1d 6816	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐽 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
57	56	fveq2d 5679	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
58	57	oveq2d 6074	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
59	54, 58	breq12d 4127	. . . . . . . . . . 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 6076	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
61	60	breq2d 4126	. . . . . . . . . . 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 4118	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (𝐽 <N 𝑘 ↔ 𝐽 <N 𝐵))
65		fveq2 5675	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐹‘𝑘) = (𝐹‘𝐵))
66	65	oveq1d 6073	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
67	66	breq2d 4126	. . . . . . . . . . 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 4124	. . . . . . . . . . 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 2937	. . . . . . . 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 7731	. . . . . . . 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 7712	. . . . . . . 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 6232	. . . . . 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 5163	. . . 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 7653	. . . 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 1344	. 2 ⊢ (𝜑 → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
88	27, 3	ffvelcdmd 5818	. . . 4 ⊢ (𝜑 → (𝐹‘𝐽) ∈ Q)
89		addclnq 7706	. . . . 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 4447	. . . . 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 697	. . 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 1004   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ⟨cop 3697   class class class wbr 4114   Or wor 4421  ⟶wf 5353  ‘cfv 5357  (class class class)co 6058  1_oc1o 6653  [cec 6778  Ncnpi 7603   <_N clti 7606   ~_Q ceq 7610  Qcnq 7611   +_Q cplq 7613  *_Qcrq 7615   <_Q cltq 7616

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-eprel 4415  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-1o 6660  df-oadd 6664  df-omul 6665  df-er 6780  df-ec 6782  df-qs 6786  df-ni 7635  df-pli 7636  df-mi 7637  df-lti 7638  df-plpq 7675  df-mpq 7676  df-enq 7678  df-nqqs 7679  df-plqqs 7680  df-mqqs 7681  df-1nqqs 7682  df-rq 7683  df-ltnqqs 7684

This theorem is referenced by:  caucvgprlemladdrl  8009