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Theorem caucvgprlemnbj 6822
Description: Lemma for caucvgpr 6837. 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 3794 . . . . . . . . . 10 (𝑛 = 𝐵 → (𝑛 <N 𝑘𝐵 <N 𝑘))
5 fveq2 5205 . . . . . . . . . . . 12 (𝑛 = 𝐵 → (𝐹𝑛) = (𝐹𝐵))
6 opeq1 3576 . . . . . . . . . . . . . . 15 (𝑛 = 𝐵 → ⟨𝑛, 1𝑜⟩ = ⟨𝐵, 1𝑜⟩)
76eceq1d 6172 . . . . . . . . . . . . . 14 (𝑛 = 𝐵 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐵, 1𝑜⟩] ~Q )
87fveq2d 5209 . . . . . . . . . . . . 13 (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))
98oveq2d 5555 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
105, 9breq12d 3804 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
115, 8oveq12d 5557 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1211breq2d 3803 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
1310, 12anbi12d 450 . . . . . . . . . 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 227 . . . . . . . . 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 3795 . . . . . . . . . 10 (𝑘 = 𝐽 → (𝐵 <N 𝑘𝐵 <N 𝐽))
16 fveq2 5205 . . . . . . . . . . . . 13 (𝑘 = 𝐽 → (𝐹𝑘) = (𝐹𝐽))
1716oveq1d 5554 . . . . . . . . . . . 12 (𝑘 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1817breq2d 3803 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
1916breq1d 3801 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
2018, 19anbi12d 450 . . . . . . . . . 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 227 . . . . . . . . 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 2684 . . . . . . . 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 397 . . . . . . 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 119 . . . . 5 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
2625simprd 111 . . . 4 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
27 caucvgpr.f . . . . . . . 8 (𝜑𝐹:NQ)
2827, 2ffvelrnd 5330 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ Q)
29 nnnq 6577 . . . . . . . 8 (𝐵N → [⟨𝐵, 1𝑜⟩] ~QQ)
30 recclnq 6547 . . . . . . . 8 ([⟨𝐵, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
312, 29, 303syl 17 . . . . . . 7 (𝜑 → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
32 addclnq 6530 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q)
3328, 31, 32syl2anc 397 . . . . . 6 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ∈ Q)
34 nnnq 6577 . . . . . . 7 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
35 recclnq 6547 . . . . . . 7 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
363, 34, 353syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
37 ltaddnq 6562 . . . . . 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 397 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
3938adantr 265 . . . 4 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
40 ltsonq 6553 . . . . 5 <Q Or Q
41 ltrelnq 6520 . . . . 5 <Q ⊆ (Q × Q)
4240, 41sotri 4747 . . . 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 397 . . 3 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
44 ltaddnq 6562 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
4528, 31, 44syl2anc 397 . . . . . 6 (𝜑 → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
4645adantr 265 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
47 fveq2 5205 . . . . . . 7 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
4847breq1d 3801 . . . . . 6 (𝐵 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
4948adantl 266 . . . . 5 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
5046, 49mpbid 139 . . . 4 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
5138adantr 265 . . . 4 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
5250, 51, 42syl2anc 397 . . 3 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
53 breq1 3794 . . . . . . . . . 10 (𝑛 = 𝐽 → (𝑛 <N 𝑘𝐽 <N 𝑘))
54 fveq2 5205 . . . . . . . . . . . 12 (𝑛 = 𝐽 → (𝐹𝑛) = (𝐹𝐽))
55 opeq1 3576 . . . . . . . . . . . . . . 15 (𝑛 = 𝐽 → ⟨𝑛, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
5655eceq1d 6172 . . . . . . . . . . . . . 14 (𝑛 = 𝐽 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
5756fveq2d 5209 . . . . . . . . . . . . 13 (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
5857oveq2d 5555 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
5954, 58breq12d 3804 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6054, 57oveq12d 5557 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6160breq2d 3803 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6259, 61anbi12d 450 . . . . . . . . . 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 227 . . . . . . . . 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 3795 . . . . . . . . . 10 (𝑘 = 𝐵 → (𝐽 <N 𝑘𝐽 <N 𝐵))
65 fveq2 5205 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐹𝑘) = (𝐹𝐵))
6665oveq1d 5554 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6766breq2d 3803 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6865breq1d 3801 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6967, 68anbi12d 450 . . . . . . . . . 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 227 . . . . . . . . 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 2684 . . . . . . . 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 397 . . . . . . 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 119 . . . . 5 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
7574simpld 109 . . . 4 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
76 ltanqg 6555 . . . . . . . 8 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
7776adantl 266 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
78 addcomnqg 6536 . . . . . . . 8 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7978adantl 266 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
8077, 28, 33, 36, 79caovord2d 5697 . . . . . 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 139 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8281adantr 265 . . . 4 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8340, 41sotri 4747 . . . 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 397 . . 3 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
85 pitri3or 6477 . . . 4 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
862, 3, 85syl2anc 397 . . 3 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
8743, 52, 84, 86mpjao3dan 1213 . 2 (𝜑 → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8827, 3ffvelrnd 5330 . . . 4 (𝜑 → (𝐹𝐽) ∈ Q)
89 addclnq 6530 . . . . 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 397 . . . 4 (𝜑 → (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
91 so2nr 4085 . . . . 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 408 . . . 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 397 . . 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 634 . . 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 141 . 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 101  wb 102  w3o 895  w3a 896   = wceq 1259  wcel 1409  wral 2323  cop 3405   class class class wbr 3791   Or wor 4059  wf 4925  cfv 4929  (class class class)co 5539  1𝑜c1o 6024  [cec 6134  Ncnpi 6427   <N clti 6430   ~Q ceq 6434  Qcnq 6435   +Q cplq 6437  *Qcrq 6439   <Q cltq 6440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508
This theorem is referenced by:  caucvgprlemladdrl  6833
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