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Theorem caucvgprlemnbj 7170
Description: Lemma for caucvgpr 7185. 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 (𝜑𝐹:NQ)
caucvgpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
caucvgprlemnbj.b (𝜑𝐵N)
caucvgprlemnbj.j (𝜑𝐽N)
Assertion
Ref Expression
caucvgprlemnbj (𝜑 → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽))
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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgpr.cau . . . . . . 7 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
2 caucvgprlemnbj.b . . . . . . . 8 (𝜑𝐵N)
3 caucvgprlemnbj.j . . . . . . . 8 (𝜑𝐽N)
4 breq1 3823 . . . . . . . . . 10 (𝑛 = 𝐵 → (𝑛 <N 𝑘𝐵 <N 𝑘))
5 fveq2 5268 . . . . . . . . . . . 12 (𝑛 = 𝐵 → (𝐹𝑛) = (𝐹𝐵))
6 opeq1 3605 . . . . . . . . . . . . . . 15 (𝑛 = 𝐵 → ⟨𝑛, 1𝑜⟩ = ⟨𝐵, 1𝑜⟩)
76eceq1d 6280 . . . . . . . . . . . . . 14 (𝑛 = 𝐵 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐵, 1𝑜⟩] ~Q )
87fveq2d 5272 . . . . . . . . . . . . 13 (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))
98oveq2d 5629 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
105, 9breq12d 3833 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
115, 8oveq12d 5631 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1211breq2d 3832 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
1310, 12anbi12d 457 . . . . . . . . . 10 (𝑛 = 𝐵 → (((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))))
144, 13imbi12d 232 . . . . . . . . 9 (𝑛 = 𝐵 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ (𝐵 <N 𝑘 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))))
15 breq2 3824 . . . . . . . . . 10 (𝑘 = 𝐽 → (𝐵 <N 𝑘𝐵 <N 𝐽))
16 fveq2 5268 . . . . . . . . . . . . 13 (𝑘 = 𝐽 → (𝐹𝑘) = (𝐹𝐽))
1716oveq1d 5628 . . . . . . . . . . . 12 (𝑘 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1817breq2d 3832 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
1916breq1d 3830 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
2018, 19anbi12d 457 . . . . . . . . . 10 (𝑘 = 𝐽 → (((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))))
2115, 20imbi12d 232 . . . . . . . . 9 (𝑘 = 𝐽 → ((𝐵 <N 𝑘 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))) ↔ (𝐵 <N 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))))
2214, 21rspc2v 2726 . . . . . . . 8 ((𝐵N𝐽N) → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) → (𝐵 <N 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))))
232, 3, 22syl2anc 403 . . . . . . 7 (𝜑 → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) → (𝐵 <N 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))))
241, 23mpd 13 . . . . . 6 (𝜑 → (𝐵 <N 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))))
2524imp 122 . . . . 5 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
2625simprd 112 . . . 4 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
27 caucvgpr.f . . . . . . . 8 (𝜑𝐹:NQ)
2827, 2ffvelrnd 5398 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ Q)
29 nnnq 6925 . . . . . . . 8 (𝐵N → [⟨𝐵, 1𝑜⟩] ~QQ)
30 recclnq 6895 . . . . . . . 8 ([⟨𝐵, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
312, 29, 303syl 17 . . . . . . 7 (𝜑 → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
32 addclnq 6878 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q)
3328, 31, 32syl2anc 403 . . . . . 6 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q)
34 nnnq 6925 . . . . . . 7 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
35 recclnq 6895 . . . . . . 7 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
363, 34, 353syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
37 ltaddnq 6910 . . . . . 6 ((((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
3833, 36, 37syl2anc 403 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
3938adantr 270 . . . 4 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
40 ltsonq 6901 . . . . 5 <Q Or Q
41 ltrelnq 6868 . . . . 5 <Q ⊆ (Q × Q)
4240, 41sotri 4794 . . . 4 (((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
4326, 39, 42syl2anc 403 . . 3 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
44 ltaddnq 6910 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
4528, 31, 44syl2anc 403 . . . . . 6 (𝜑 → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
4645adantr 270 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
47 fveq2 5268 . . . . . . 7 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
4847breq1d 3830 . . . . . 6 (𝐵 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
4948adantl 271 . . . . 5 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
5046, 49mpbid 145 . . . 4 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
5138adantr 270 . . . 4 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
5250, 51, 42syl2anc 403 . . 3 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
53 breq1 3823 . . . . . . . . . 10 (𝑛 = 𝐽 → (𝑛 <N 𝑘𝐽 <N 𝑘))
54 fveq2 5268 . . . . . . . . . . . 12 (𝑛 = 𝐽 → (𝐹𝑛) = (𝐹𝐽))
55 opeq1 3605 . . . . . . . . . . . . . . 15 (𝑛 = 𝐽 → ⟨𝑛, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
5655eceq1d 6280 . . . . . . . . . . . . . 14 (𝑛 = 𝐽 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
5756fveq2d 5272 . . . . . . . . . . . . 13 (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
5857oveq2d 5629 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
5954, 58breq12d 3833 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6054, 57oveq12d 5631 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6160breq2d 3832 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6259, 61anbi12d 457 . . . . . . . . . 10 (𝑛 = 𝐽 → (((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
6353, 62imbi12d 232 . . . . . . . . 9 (𝑛 = 𝐽 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ (𝐽 <N 𝑘 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
64 breq2 3824 . . . . . . . . . 10 (𝑘 = 𝐵 → (𝐽 <N 𝑘𝐽 <N 𝐵))
65 fveq2 5268 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐹𝑘) = (𝐹𝐵))
6665oveq1d 5628 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6766breq2d 3832 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6865breq1d 3830 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6967, 68anbi12d 457 . . . . . . . . . 10 (𝑘 = 𝐵 → (((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
7064, 69imbi12d 232 . . . . . . . . 9 (𝑘 = 𝐵 → ((𝐽 <N 𝑘 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))) ↔ (𝐽 <N 𝐵 → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
7163, 70rspc2v 2726 . . . . . . . 8 ((𝐽N𝐵N) → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) → (𝐽 <N 𝐵 → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
723, 2, 71syl2anc 403 . . . . . . 7 (𝜑 → (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) → (𝐽 <N 𝐵 → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
731, 72mpd 13 . . . . . 6 (𝜑 → (𝐽 <N 𝐵 → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
7473imp 122 . . . . 5 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
7574simpld 110 . . . 4 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
76 ltanqg 6903 . . . . . . . 8 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
7776adantl 271 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
78 addcomnqg 6884 . . . . . . . 8 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7978adantl 271 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
8077, 28, 33, 36, 79caovord2d 5771 . . . . . 6 (𝜑 → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
8145, 80mpbid 145 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8281adantr 270 . . . 4 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8340, 41sotri 4794 . . . 4 (((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8475, 82, 83syl2anc 403 . . 3 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
85 pitri3or 6825 . . . 4 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
862, 3, 85syl2anc 403 . . 3 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
8743, 52, 84, 86mpjao3dan 1241 . 2 (𝜑 → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8827, 3ffvelrnd 5398 . . . 4 (𝜑 → (𝐹𝐽) ∈ Q)
89 addclnq 6878 . . . . 5 ((((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
9033, 36, 89syl2anc 403 . . . 4 (𝜑 → (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
91 so2nr 4122 . . . . 5 (( <Q Or Q ∧ ((𝐹𝐽) ∈ Q ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)) → ¬ ((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
9240, 91mpan 415 . . . 4 (((𝐹𝐽) ∈ Q ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q) → ¬ ((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
9388, 90, 92syl2anc 403 . . 3 (𝜑 → ¬ ((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
94 imnan 657 . . 3 (((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)) ↔ ¬ ((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
9593, 94sylibr 132 . 2 (𝜑 → ((𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
9687, 95mpd 13 1 (𝜑 → ¬ (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3o 921  w3a 922   = wceq 1287  wcel 1436  wral 2355  cop 3434   class class class wbr 3820   Or wor 4096  wf 4977  cfv 4981  (class class class)co 5613  1𝑜c1o 6128  [cec 6242  Ncnpi 6775   <N clti 6778   ~Q ceq 6782  Qcnq 6783   +Q cplq 6785  *Qcrq 6787   <Q cltq 6788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3929  ax-sep 3932  ax-nul 3940  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-iinf 4376
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-int 3672  df-iun 3715  df-br 3821  df-opab 3875  df-mpt 3876  df-tr 3912  df-eprel 4090  df-id 4094  df-po 4097  df-iso 4098  df-iord 4167  df-on 4169  df-suc 4172  df-iom 4379  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-f1 4986  df-fo 4987  df-f1o 4988  df-fv 4989  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-1st 5868  df-2nd 5869  df-recs 6024  df-irdg 6089  df-1o 6135  df-oadd 6139  df-omul 6140  df-er 6244  df-ec 6246  df-qs 6250  df-ni 6807  df-pli 6808  df-mi 6809  df-lti 6810  df-plpq 6847  df-mpq 6848  df-enq 6850  df-nqqs 6851  df-plqqs 6852  df-mqqs 6853  df-1nqqs 6854  df-rq 6855  df-ltnqqs 6856
This theorem is referenced by:  caucvgprlemladdrl  7181
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