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Theorem caucvgprlemnbj 7129
Description: Lemma for caucvgpr 7144. 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 3814 . . . . . . . . . 10 (𝑛 = 𝐵 → (𝑛 <N 𝑘𝐵 <N 𝑘))
5 fveq2 5253 . . . . . . . . . . . 12 (𝑛 = 𝐵 → (𝐹𝑛) = (𝐹𝐵))
6 opeq1 3596 . . . . . . . . . . . . . . 15 (𝑛 = 𝐵 → ⟨𝑛, 1𝑜⟩ = ⟨𝐵, 1𝑜⟩)
76eceq1d 6258 . . . . . . . . . . . . . 14 (𝑛 = 𝐵 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐵, 1𝑜⟩] ~Q )
87fveq2d 5257 . . . . . . . . . . . . 13 (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))
98oveq2d 5607 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
105, 9breq12d 3824 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
115, 8oveq12d 5609 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1211breq2d 3823 . . . . . . . . . . 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 3815 . . . . . . . . . 10 (𝑘 = 𝐽 → (𝐵 <N 𝑘𝐵 <N 𝐽))
16 fveq2 5253 . . . . . . . . . . . . 13 (𝑘 = 𝐽 → (𝐹𝑘) = (𝐹𝐽))
1716oveq1d 5606 . . . . . . . . . . . 12 (𝑘 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
1817breq2d 3823 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ))))
1916breq1d 3821 . . . . . . . . . . 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 2723 . . . . . . . 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 5380 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ Q)
29 nnnq 6884 . . . . . . . 8 (𝐵N → [⟨𝐵, 1𝑜⟩] ~QQ)
30 recclnq 6854 . . . . . . . 8 ([⟨𝐵, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
312, 29, 303syl 17 . . . . . . 7 (𝜑 → (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ∈ Q)
32 addclnq 6837 . . . . . . 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 6884 . . . . . . 7 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
35 recclnq 6854 . . . . . . 7 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
363, 34, 353syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
37 ltaddnq 6869 . . . . . 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 6860 . . . . 5 <Q Or Q
41 ltrelnq 6827 . . . . 5 <Q ⊆ (Q × Q)
4240, 41sotri 4782 . . . 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 6869 . . . . . . 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 5253 . . . . . . 7 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
4847breq1d 3821 . . . . . 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 3814 . . . . . . . . . 10 (𝑛 = 𝐽 → (𝑛 <N 𝑘𝐽 <N 𝑘))
54 fveq2 5253 . . . . . . . . . . . 12 (𝑛 = 𝐽 → (𝐹𝑛) = (𝐹𝐽))
55 opeq1 3596 . . . . . . . . . . . . . . 15 (𝑛 = 𝐽 → ⟨𝑛, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
5655eceq1d 6258 . . . . . . . . . . . . . 14 (𝑛 = 𝐽 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
5756fveq2d 5257 . . . . . . . . . . . . 13 (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
5857oveq2d 5607 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
5954, 58breq12d 3824 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6054, 57oveq12d 5609 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6160breq2d 3823 . . . . . . . . . . 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 3815 . . . . . . . . . 10 (𝑘 = 𝐵 → (𝐽 <N 𝑘𝐽 <N 𝐵))
65 fveq2 5253 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐹𝑘) = (𝐹𝐵))
6665oveq1d 5606 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6766breq2d 3823 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
6865breq1d 3821 . . . . . . . . . . 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 2723 . . . . . . . 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 6862 . . . . . . . 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 6843 . . . . . . . 8 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7978adantl 271 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
8077, 28, 33, 36, 79caovord2d 5749 . . . . . 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 4782 . . . 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 6784 . . . 4 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
862, 3, 85syl2anc 403 . . 3 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
8743, 52, 84, 86mpjao3dan 1239 . 2 (𝜑 → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
8827, 3ffvelrnd 5380 . . . 4 (𝜑 → (𝐹𝐽) ∈ Q)
89 addclnq 6837 . . . . 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 4112 . . . . 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 919  w3a 920   = wceq 1285  wcel 1434  wral 2353  cop 3425   class class class wbr 3811   Or wor 4086  wf 4965  cfv 4969  (class class class)co 5591  1𝑜c1o 6106  [cec 6220  Ncnpi 6734   <N clti 6737   ~Q ceq 6741  Qcnq 6742   +Q cplq 6744  *Qcrq 6746   <Q cltq 6747
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-eprel 4080  df-id 4084  df-po 4087  df-iso 4088  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-1o 6113  df-oadd 6117  df-omul 6118  df-er 6222  df-ec 6224  df-qs 6228  df-ni 6766  df-pli 6767  df-mi 6768  df-lti 6769  df-plpq 6806  df-mpq 6807  df-enq 6809  df-nqqs 6810  df-plqqs 6811  df-mqqs 6812  df-1nqqs 6813  df-rq 6814  df-ltnqqs 6815
This theorem is referenced by:  caucvgprlemladdrl  7140
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