Theorem caucvgprlemnbj 7734

Description: Lemma for caucvgpr 7749. 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 4036	. . . . . . . . . 10 ⊢ (𝑛 = 𝐵 → (𝑛 <N 𝑘 ↔ 𝐵 <N 𝑘))
5		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → (𝐹‘𝑛) = (𝐹‘𝐵))
6		opeq1 3808	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐵 → ⟨𝑛, 1o⟩ = ⟨𝐵, 1o⟩)
7	6	eceq1d 6628	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐵 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐵, 1o⟩] ~Q )
8	7	fveq2d 5562	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐵, 1o⟩] ~Q ))
9	8	oveq2d 5938	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
10	5, 9	breq12d 4046	. . . . . . . . . . 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 5940	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐵 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
12	11	breq2d 4045	. . . . . . . . . . 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 4037	. . . . . . . . . 10 ⊢ (𝑘 = 𝐽 → (𝐵 <N 𝑘 ↔ 𝐵 <N 𝐽))
16		fveq2 5558	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐽 → (𝐹‘𝑘) = (𝐹‘𝐽))
17	16	oveq1d 5937	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
18	17	breq2d 4045	. . . . . . . . . . 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 4043	. . . . . . . . . . 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 2881	. . . . . . . 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 5698	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐵) ∈ Q)
29		nnnq 7489	. . . . . . . 8 ⊢ (𝐵 ∈ N → [⟨𝐵, 1o⟩] ~Q ∈ Q)
30		recclnq 7459	. . . . . . . 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 7442	. . . . . . 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 7489	. . . . . . 7 ⊢ (𝐽 ∈ N → [⟨𝐽, 1o⟩] ~Q ∈ Q)
35		recclnq 7459	. . . . . . 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 7474	. . . . . 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 7465	. . . . 5 ⊢ <Q Or Q
41		ltrelnq 7432	. . . . 5 ⊢ <Q ⊆ (Q × Q)
42	40, 41	sotri 5065	. . . 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 7474	. . . . . . 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 5558	. . . . . . 7 ⊢ (𝐵 = 𝐽 → (𝐹‘𝐵) = (𝐹‘𝐽))
48	47	breq1d 4043	. . . . . 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 4036	. . . . . . . . . 10 ⊢ (𝑛 = 𝐽 → (𝑛 <N 𝑘 ↔ 𝐽 <N 𝑘))
54		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → (𝐹‘𝑛) = (𝐹‘𝐽))
55		opeq1 3808	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝐽 → ⟨𝑛, 1o⟩ = ⟨𝐽, 1o⟩)
56	55	eceq1d 6628	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝐽 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
57	56	fveq2d 5562	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
58	57	oveq2d 5938	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
59	54, 58	breq12d 4046	. . . . . . . . . . 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 5940	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝐽 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
61	60	breq2d 4045	. . . . . . . . . . 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 4037	. . . . . . . . . 10 ⊢ (𝑘 = 𝐵 → (𝐽 <N 𝑘 ↔ 𝐽 <N 𝐵))
65		fveq2 5558	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐵 → (𝐹‘𝑘) = (𝐹‘𝐵))
66	65	oveq1d 5937	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐵 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) = ((𝐹‘𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
67	66	breq2d 4045	. . . . . . . . . . 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 4043	. . . . . . . . . . 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 2881	. . . . . . . 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 7467	. . . . . . . 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 7448	. . . . . . . 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 6093	. . . . . 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 5065	. . . 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 7389	. . . 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 1318	. 2 ⊢ (𝜑 → (𝐹‘𝐽) <Q (((𝐹‘𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
88	27, 3	ffvelcdmd 5698	. . . 4 ⊢ (𝜑 → (𝐹‘𝐽) ∈ Q)
89		addclnq 7442	. . . . 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 4356	. . . . 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 691	. . 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 979   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⟨cop 3625   class class class wbr 4033   Or wor 4330  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1_oc1o 6467  [cec 6590  Ncnpi 7339   <_N clti 7342   ~_Q ceq 7346  Qcnq 7347   +_Q cplq 7349  *_Qcrq 7351   <_Q cltq 7352

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420

This theorem is referenced by:  caucvgprlemladdrl  7745