Theorem caucvgprlemnbj 8028

Description: Lemma for caucvgpr 8043. 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 4131	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → (𝑛 <N 𝑘 ↔ 𝐵 <N 𝑘))
5		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → (𝐹‘𝑛) = (𝐹‘𝐵))
6		opeq1 3902	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐵 → ⟨𝑛, 1o⟩ = ⟨𝐵, 1o⟩)
7	6	eceq1d 6837	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐵 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐵, 1o⟩] ~Q )
8	7	fveq2d 5697	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐵, 1o⟩] ~Q ))
9	8	oveq2d 6095	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
10	5, 9	breq12d 4141	. . . . . . . . . . 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 6097	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
12	11	breq2d 4140	. . . . . . . . . . 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 477	. . . . . . . . . 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 4132	. . . . . . . . . 10 ⊢ (𝑘 = 𝐽 → (𝐵 <N 𝑘 ↔ 𝐵 <N 𝐽))
16		fveq2 5693	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
17	16	oveq1d 6094	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
18	17	breq2d 4140	. . . . . . . . . . 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 4138	. . . . . . . . . . 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 477	. . . . . . . . . 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 2943	. . . . . . . 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 415	. . . . . . 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 5838	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵) ∈ Q)
29		nnnq 7783	. . . . . . . 8 ⊢ (𝐵 ∈ N → [⟨𝐵, 1o⟩] ~Q ∈ Q)
30		recclnq 7753	. . . . . . . 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 7736	. . . . . . 7 ⊢ (((𝐹‘𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
33	28, 31, 32	syl2anc 415	. . . . . 6 ⊢ (𝜑 → ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
34		nnnq 7783	. . . . . . 7 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
35		recclnq 7753	. . . . . . 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 7768	. . . . . 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 415	. . . . 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 7759	. . . . 5 ⊢ <Q Or Q
41		ltrelnq 7726	. . . . 5 ⊢ <Q ⊆ (Q × Q)
42	40, 41	sotri 5181	. . . 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 415	. . 3 ⊢ ((𝜑 ∧ 𝐵 <N 𝐽) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
44		ltaddnq 7768	. . . . . . 7 ⊢ (((𝐹‘𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → (𝐹‘𝐵) <Q ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
45	28, 31, 44	syl2anc 415	. . . . . 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 5693	. . . . . . 7 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
48	47	breq1d 4138	. . . . . 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 415	. . 3 ⊢ ((𝜑 ∧ 𝐵 = 𝐽) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
53		breq1 4131	. . . . . . . . . 10 ⊢ (𝑛 = 𝐽 → (𝑛 <N 𝑘 ↔ 𝐽 <N 𝑘))
54		fveq2 5693	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (𝐹‘𝑛) = (𝐹‘𝐽))
55		opeq1 3902	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐽 → ⟨𝑛, 1o⟩ = ⟨𝐽, 1o⟩)
56	55	eceq1d 6837	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐽 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
57	56	fveq2d 5697	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
58	57	oveq2d 6095	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
59	54, 58	breq12d 4141	. . . . . . . . . . 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 6097	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
61	60	breq2d 4140	. . . . . . . . . . 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 477	. . . . . . . . . 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 4132	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (𝐽 <N 𝑘 ↔ 𝐽 <N 𝐵))
65		fveq2 5693	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐹‘𝑘) = (𝐹‘𝐵))
66	65	oveq1d 6094	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
67	66	breq2d 4140	. . . . . . . . . . 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 4138	. . . . . . . . . . 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 477	. . . . . . . . . 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 2943	. . . . . . . 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 415	. . . . . . 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 7761	. . . . . . . 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 7742	. . . . . . . 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 6253	. . . . . 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 5181	. . . 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 415	. . 3 ⊢ ((𝜑 ∧ 𝐽 <N 𝐵) → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
85		pitri3or 7683	. . . 4 ⊢ ((𝐵 ∈ N ∧ 𝐽 ∈ N) → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
86	2, 3, 85	syl2anc 415	. . 3 ⊢ (𝜑 → (𝐵 <N 𝐽 ∨ 𝐵 = 𝐽 ∨ 𝐽 <N 𝐵))
87	43, 52, 84, 86	mpjao3dan 1348	. 2 ⊢ (𝜑 → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
88	27, 3	ffvelcdmd 5838	. . . 4 ⊢ (𝜑 → (𝐹‘𝐽) ∈ Q)
89		addclnq 7736	. . . . 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 415	. . . 4 ⊢ (𝜑 → (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q)
91		so2nr 4464	. . . . 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 428	. . . 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 415	. . 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 701	. . 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 1008   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ⟨cop 3711   class class class wbr 4128   Or wor 4438  ⟶wf 5371  ‘cfv 5375  (class class class)co 6079  1_oc1o 6674  [cec 6799  Ncnpi 7633   <_N clti 7636   ~_Q ceq 7640  Qcnq 7641   +_Q cplq 7643  *_Qcrq 7645   <_Q cltq 7646

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-eprel 4432  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-1o 6681  df-oadd 6685  df-omul 6686  df-er 6801  df-ec 6803  df-qs 6807  df-ni 7665  df-pli 7666  df-mi 7667  df-lti 7668  df-plpq 7705  df-mpq 7706  df-enq 7708  df-nqqs 7709  df-plqqs 7710  df-mqqs 7711  df-1nqqs 7712  df-rq 7713  df-ltnqqs 7714

This theorem is referenced by:  caucvgprlemladdrl  8039