Theorem caucvgprlemnkj 6764

Description: Lemma for caucvgpr 6780. Part of disjointness. (Contributed by Jim Kingdon, 23-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 )))))
caucvgprlemnkj.k	⊢ (𝜑 → 𝐾 ∈ N)
caucvgprlemnkj.j	⊢ (𝜑 → 𝐽 ∈ N)
caucvgprlemnkj.s	⊢ (𝜑 → 𝑆 ∈ Q)

Assertion

Ref	Expression
caucvgprlemnkj	⊢ (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))

Distinct variable group:   𝑘,𝐹,𝑛

Allowed substitution hints:   𝜑(𝑘,𝑛)   𝑆(𝑘,𝑛)   𝐽(𝑘,𝑛)   𝐾(𝑘,𝑛)

Proof of Theorem caucvgprlemnkj

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

Step	Hyp	Ref	Expression
1		ltsonq 6496	. . . 4 ⊢ <Q Or Q
2		ltrelnq 6463	. . . 4 ⊢ <Q ⊆ (Q × Q)
3	1, 2	son2lpi 4721	. . 3 ⊢ ¬ (𝑆 <Q (𝐹‘𝐽) ∧ (𝐹‘𝐽) <Q 𝑆)
4		simprl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾))
5		caucvgpr.cau	. . . . . . . . . . . 12 ⊢ (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 )))))
6		breq1 3767	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑎 → (𝑛 <N 𝑘 ↔ 𝑎 <N 𝑘))
7		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → (𝐹‘𝑛) = (𝐹‘𝑎))
8		opeq1 3549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑎 → ⟨𝑛, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
9	8	eceq1d 6142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
10	9	fveq2d 5182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
11	10	oveq2d 5528	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
12	7, 11	breq12d 3777	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
13	7, 10	oveq12d 5530	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
14	13	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑎 → ((𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
15	12, 14	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑎 → (((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
16	6, 15	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑎 → ((𝑛 <N 𝑘 → ((𝐹‘𝑛) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ (𝑎 <N 𝑘 → ((𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))))
17		breq2 3768	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑏 → (𝑎 <N 𝑘 ↔ 𝑎 <N 𝑏))
18		fveq2 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑏 → (𝐹‘𝑘) = (𝐹‘𝑏))
19	18	oveq1d 5527	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
20	19	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
21	18	breq1d 3774	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑏 → ((𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
22	20, 21	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑏 → (((𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
23	17, 22	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑏 → ((𝑎 <N 𝑘 → ((𝐹‘𝑎) <Q ((𝐹‘𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑘) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) ↔ (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))))
24	16, 23	cbvral2v 2541	. . . . . . . . . . . 12 ⊢ (∀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 )))) ↔ ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
25	5, 24	sylib 127	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
26		caucvgprlemnkj.k	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ N)
27		caucvgprlemnkj.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ N)
28		breq1 3767	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐾 → (𝑎 <N 𝑏 ↔ 𝐾 <N 𝑏))
29		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐾 → (𝐹‘𝑎) = (𝐹‘𝐾))
30		opeq1 3549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐾 → ⟨𝑎, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
31	30	eceq1d 6142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐾 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
32	31	fveq2d 5182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐾 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
33	32	oveq2d 5528	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐾 → ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
34	29, 33	breq12d 3777	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐾 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
35	29, 32	oveq12d 5530	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐾 → ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
36	35	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐾 → ((𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
37	34, 36	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐾 → (((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
38	28, 37	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐾 → ((𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) ↔ (𝐾 <N 𝑏 → ((𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
39		breq2 3768	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐽 → (𝐾 <N 𝑏 ↔ 𝐾 <N 𝐽))
40		fveq2 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝐽 → (𝐹‘𝑏) = (𝐹‘𝐽))
41	40	oveq1d 5527	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐽 → ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
42	41	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐽 → ((𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
43	40	breq1d 3774	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐽 → ((𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
44	42, 43	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐽 → (((𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
45	39, 44	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐽 → ((𝐾 <N 𝑏 → ((𝐹‘𝐾) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))) ↔ (𝐾 <N 𝐽 → ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
46	38, 45	rspc2v 2662	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ N ∧ 𝐽 ∈ N) → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐾 <N 𝐽 → ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
47	26, 27, 46	syl2anc 391	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐾 <N 𝐽 → ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
48	25, 47	mpd 13	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾 <N 𝐽 → ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
49	48	imp 115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ((𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
50	49	simpld 105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
51	50	adantr 261	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
52	1, 2	sotri 4720	. . . . . . 7 ⊢ (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
53	4, 51, 52	syl2anc 391	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
54		ltanqg 6498	. . . . . . . 8 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
55	54	adantl 262	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
56		caucvgprlemnkj.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ Q)
57	56	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 ∈ Q)
58		caucvgpr.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:N⟶Q)
59	58, 27	ffvelrnd 5303	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘𝐽) ∈ Q)
60	59	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹‘𝐽) ∈ Q)
61		nnnq 6520	. . . . . . . . 9 ⊢ (𝐾 ∈ N → [⟨𝐾, 1𝑜⟩] ~Q ∈ Q)
62		recclnq 6490	. . . . . . . . 9 ⊢ ([⟨𝐾, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
63	26, 61, 62	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
64	63	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
65		addcomnqg 6479	. . . . . . . 8 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
66	65	adantl 262	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
67	55, 57, 60, 64, 66	caovord2d 5670	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q (𝐹‘𝐽) ↔ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
68	53, 67	mpbird 156	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q (𝐹‘𝐽))
69		nnnq 6520	. . . . . . . . 9 ⊢ (𝐽 ∈ N → [⟨𝐽, 1𝑜⟩] ~Q ∈ Q)
70		recclnq 6490	. . . . . . . . 9 ⊢ ([⟨𝐽, 1𝑜⟩] ~Q ∈ Q → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
71	27, 69, 70	3syl 17	. . . . . . . 8 ⊢ (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
72	71	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
73		ltaddnq 6505	. . . . . . 7 ⊢ (((𝐹‘𝐽) ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (𝐹‘𝐽) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
74	60, 72, 73	syl2anc 391	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹‘𝐽) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
75		simprr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
76	1, 2	sotri 4720	. . . . . 6 ⊢ (((𝐹‘𝐽) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝐹‘𝐽) <Q 𝑆)
77	74, 75, 76	syl2anc 391	. . . . 5 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹‘𝐽) <Q 𝑆)
78	68, 77	jca 290	. . . 4 ⊢ (((𝜑 ∧ 𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q (𝐹‘𝐽) ∧ (𝐹‘𝐽) <Q 𝑆))
79	78	ex 108	. . 3 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝑆 <Q (𝐹‘𝐽) ∧ (𝐹‘𝐽) <Q 𝑆)))
80	3, 79	mtoi 590	. 2 ⊢ ((𝜑 ∧ 𝐾 <N 𝐽) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
81	1, 2	son2lpi 4721	. . 3 ⊢ ¬ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆 ∧ 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
82		opeq1 3549	. . . . . . . . . . . 12 ⊢ (𝐾 = 𝐽 → ⟨𝐾, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
83	82	eceq1d 6142	. . . . . . . . . . 11 ⊢ (𝐾 = 𝐽 → [⟨𝐾, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
84	83	fveq2d 5182	. . . . . . . . . 10 ⊢ (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
85	84	oveq2d 5528	. . . . . . . . 9 ⊢ (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
86		fveq2 5178	. . . . . . . . 9 ⊢ (𝐾 = 𝐽 → (𝐹‘𝐾) = (𝐹‘𝐽))
87	85, 86	breq12d 3777	. . . . . . . 8 ⊢ (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽)))
88	87	anbi1d 438	. . . . . . 7 ⊢ (𝐾 = 𝐽 → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
89	88	adantl 262	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
90	54	adantl 262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q)) → (𝑓 <Q 𝑔 ↔ (ℎ +Q 𝑓) <Q (ℎ +Q 𝑔)))
91		addclnq 6473	. . . . . . . . . 10 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
92	56, 71, 91	syl2anc 391	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
93	65	adantl 262	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ Q ∧ 𝑔 ∈ Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
94	90, 92, 59, 71, 93	caovord2d 5670	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
95	94	adantr 261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
96	95	anbi1d 438	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐽) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
97	89, 96	bitrd 177	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
98	1, 2	sotri 4720	. . . . 5 ⊢ ((((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
99	97, 98	syl6bi 152	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
100		ltaddnq 6505	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
101	56, 71, 100	syl2anc 391	. . . . . 6 ⊢ (𝜑 → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
102		ltaddnq 6505	. . . . . . 7 ⊢ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
103	92, 71, 102	syl2anc 391	. . . . . 6 ⊢ (𝜑 → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
104	1, 2	sotri 4720	. . . . . 6 ⊢ ((𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
105	101, 103, 104	syl2anc 391	. . . . 5 ⊢ (𝜑 → 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
106	105	adantr 261	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
107	99, 106	jctird 300	. . 3 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆 ∧ 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
108	81, 107	mtoi 590	. 2 ⊢ ((𝜑 ∧ 𝐾 = 𝐽) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
109	1, 2	son2lpi 4721	. . 3 ⊢ ¬ (𝑆 <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
110	56	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 ∈ Q)
111	63	ad2antrr 457	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
112		ltaddnq 6505	. . . . . . 7 ⊢ ((𝑆 ∈ Q ∧ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
113	110, 111, 112	syl2anc 391	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
114		simprl 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾))
115		breq1 3767	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐽 → (𝑎 <N 𝑏 ↔ 𝐽 <N 𝑏))
116		fveq2 5178	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐽 → (𝐹‘𝑎) = (𝐹‘𝐽))
117		opeq1 3549	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐽 → ⟨𝑎, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
118	117	eceq1d 6142	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐽 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
119	118	fveq2d 5182	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = 𝐽 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
120	119	oveq2d 5528	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐽 → ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
121	116, 120	breq12d 3777	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐽 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
122	116, 119	oveq12d 5530	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = 𝐽 → ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
123	122	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝐽 → ((𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝑏) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
124	121, 123	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝐽 → (((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝐽) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
125	115, 124	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐽 → ((𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 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 ))))))
126		breq2 3768	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (𝐽 <N 𝑏 ↔ 𝐽 <N 𝐾))
127		fveq2 5178	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = 𝐾 → (𝐹‘𝑏) = (𝐹‘𝐾))
128	127	oveq1d 5527	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝐾 → ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) = ((𝐹‘𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
129	128	breq2d 3776	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐾 → ((𝐹‘𝐽) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
130	127	breq1d 3774	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝐾 → ((𝐹‘𝑏) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
131	129, 130	anbi12d 442	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝐾 → (((𝐹‘𝐽) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) ↔ ((𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
132	126, 131	imbi12d 223	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝐾 → ((𝐽 <N 𝑏 → ((𝐹‘𝐽) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 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 ))))))
133	125, 132	rspc2v 2662	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 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 ))))))
134	27, 26, 133	syl2anc 391	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎) <Q ((𝐹‘𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝑏) <Q ((𝐹‘𝑎) +Q (*Q‘[⟨𝑎, 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 ))))))
135	25, 134	mpd 13	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 <N 𝐾 → ((𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
136	135	imp 115	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ((𝐹‘𝐽) <Q ((𝐹‘𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
137	136	simprd 107	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
138	137	adantr 261	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
139	1, 2	sotri 4720	. . . . . . 7 ⊢ (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ (𝐹‘𝐾) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
140	114, 138, 139	syl2anc 391	. . . . . 6 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
141	1, 2	sotri 4720	. . . . . 6 ⊢ ((𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → 𝑆 <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
142	113, 140, 141	syl2anc 391	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
143		simprr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
144	142, 143	jca 290	. . . 4 ⊢ (((𝜑 ∧ 𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
145	144	ex 108	. . 3 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝑆 <Q ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
146	109, 145	mtoi 590	. 2 ⊢ ((𝜑 ∧ 𝐽 <N 𝐾) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
147		pitri3or 6420	. . 3 ⊢ ((𝐾 ∈ N ∧ 𝐽 ∈ N) → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾))
148	26, 27, 147	syl2anc 391	. 2 ⊢ (𝜑 → (𝐾 <N 𝐽 ∨ 𝐾 = 𝐽 ∨ 𝐽 <N 𝐾))
149	80, 108, 146, 148	mpjao3dan 1202	1 ⊢ (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹‘𝐾) ∧ ((𝐹‘𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ w3o 884   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ⟨cop 3378   class class class wbr 3764  ⟶wf 4898  ‘cfv 4902  (class class class)co 5512  1_𝑜c1o 5994  [cec 6104  Ncnpi 6370   <_N clti 6373   ~_Q ceq 6377  Qcnq 6378   +_Q cplq 6380  *_Qcrq 6382   <_Q cltq 6383

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451

This theorem is referenced by:  caucvgprlemdisj  6772