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Theorem caucvgprlemnbj 7495
 Description: Lemma for caucvgpr 7510. 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 3936 . . . . . . . . . 10 (𝑛 = 𝐵 → (𝑛 <N 𝑘𝐵 <N 𝑘))
5 fveq2 5425 . . . . . . . . . . . 12 (𝑛 = 𝐵 → (𝐹𝑛) = (𝐹𝐵))
6 opeq1 3709 . . . . . . . . . . . . . . 15 (𝑛 = 𝐵 → ⟨𝑛, 1o⟩ = ⟨𝐵, 1o⟩)
76eceq1d 6469 . . . . . . . . . . . . . 14 (𝑛 = 𝐵 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐵, 1o⟩] ~Q )
87fveq2d 5429 . . . . . . . . . . . . 13 (𝑛 = 𝐵 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐵, 1o⟩] ~Q ))
98oveq2d 5794 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
105, 9breq12d 3946 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
115, 8oveq12d 5796 . . . . . . . . . . . 12 (𝑛 = 𝐵 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
1211breq2d 3945 . . . . . . . . . . 11 (𝑛 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
1310, 12anbi12d 465 . . . . . . . . . 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 233 . . . . . . . . 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 3937 . . . . . . . . . 10 (𝑘 = 𝐽 → (𝐵 <N 𝑘𝐵 <N 𝐽))
16 fveq2 5425 . . . . . . . . . . . . 13 (𝑘 = 𝐽 → (𝐹𝑘) = (𝐹𝐽))
1716oveq1d 5793 . . . . . . . . . . . 12 (𝑘 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
1817breq2d 3945 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
1916breq1d 3943 . . . . . . . . . . 11 (𝑘 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
2018, 19anbi12d 465 . . . . . . . . . 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 233 . . . . . . . . 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 2803 . . . . . . . 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 409 . . . . . . 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 123 . . . . 5 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
2625simprd 113 . . . 4 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
27 caucvgpr.f . . . . . . . 8 (𝜑𝐹:NQ)
2827, 2ffvelrnd 5560 . . . . . . 7 (𝜑 → (𝐹𝐵) ∈ Q)
29 nnnq 7250 . . . . . . . 8 (𝐵N → [⟨𝐵, 1o⟩] ~QQ)
30 recclnq 7220 . . . . . . . 8 ([⟨𝐵, 1o⟩] ~QQ → (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q)
312, 29, 303syl 17 . . . . . . 7 (𝜑 → (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q)
32 addclnq 7203 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
3328, 31, 32syl2anc 409 . . . . . 6 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ∈ Q)
34 nnnq 7250 . . . . . . 7 (𝐽N → [⟨𝐽, 1o⟩] ~QQ)
35 recclnq 7220 . . . . . . 7 ([⟨𝐽, 1o⟩] ~QQ → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
363, 34, 353syl 17 . . . . . 6 (𝜑 → (*Q‘[⟨𝐽, 1o⟩] ~Q ) ∈ Q)
37 ltaddnq 7235 . . . . . 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 409 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
3938adantr 274 . . . 4 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
40 ltsonq 7226 . . . . 5 <Q Or Q
41 ltrelnq 7193 . . . . 5 <Q ⊆ (Q × Q)
4240, 41sotri 4938 . . . 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 409 . . 3 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
44 ltaddnq 7235 . . . . . . 7 (((𝐹𝐵) ∈ Q ∧ (*Q‘[⟨𝐵, 1o⟩] ~Q ) ∈ Q) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
4528, 31, 44syl2anc 409 . . . . . 6 (𝜑 → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
4645adantr 274 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
47 fveq2 5425 . . . . . . 7 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
4847breq1d 3943 . . . . . 6 (𝐵 = 𝐽 → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
4948adantl 275 . . . . 5 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q ))))
5046, 49mpbid 146 . . . 4 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
5138adantr 274 . . . 4 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
5250, 51, 42syl2anc 409 . . 3 ((𝜑𝐵 = 𝐽) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
53 breq1 3936 . . . . . . . . . 10 (𝑛 = 𝐽 → (𝑛 <N 𝑘𝐽 <N 𝑘))
54 fveq2 5425 . . . . . . . . . . . 12 (𝑛 = 𝐽 → (𝐹𝑛) = (𝐹𝐽))
55 opeq1 3709 . . . . . . . . . . . . . . 15 (𝑛 = 𝐽 → ⟨𝑛, 1o⟩ = ⟨𝐽, 1o⟩)
5655eceq1d 6469 . . . . . . . . . . . . . 14 (𝑛 = 𝐽 → [⟨𝑛, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
5756fveq2d 5429 . . . . . . . . . . . . 13 (𝑛 = 𝐽 → (*Q‘[⟨𝑛, 1o⟩] ~Q ) = (*Q‘[⟨𝐽, 1o⟩] ~Q ))
5857oveq2d 5794 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
5954, 58breq12d 3946 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
6054, 57oveq12d 5796 . . . . . . . . . . . 12 (𝑛 = 𝐽 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
6160breq2d 3945 . . . . . . . . . . 11 (𝑛 = 𝐽 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1o⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
6259, 61anbi12d 465 . . . . . . . . . 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 233 . . . . . . . . 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 3937 . . . . . . . . . 10 (𝑘 = 𝐵 → (𝐽 <N 𝑘𝐽 <N 𝐵))
65 fveq2 5425 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐹𝑘) = (𝐹𝐵))
6665oveq1d 5793 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) = ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
6766breq2d 3945 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝐽) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
6865breq1d 3943 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐹𝑘) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ↔ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
6967, 68anbi12d 465 . . . . . . . . . 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 233 . . . . . . . . 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 2803 . . . . . . . 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 409 . . . . . . 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 123 . . . . 5 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∧ (𝐹𝐵) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q ))))
7574simpld 111 . . . 4 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
76 ltanqg 7228 . . . . . . . 8 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
7776adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
78 addcomnqg 7209 . . . . . . . 8 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
7978adantl 275 . . . . . . 7 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
8077, 28, 33, 36, 79caovord2d 5944 . . . . . 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 146 . . . . 5 (𝜑 → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
8281adantr 274 . . . 4 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐵) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
8340, 41sotri 4938 . . . 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 409 . . 3 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
85 pitri3or 7150 . . . 4 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
862, 3, 85syl2anc 409 . . 3 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
8743, 52, 84, 86mpjao3dan 1286 . 2 (𝜑 → (𝐹𝐽) <Q (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
8827, 3ffvelrnd 5560 . . . 4 (𝜑 → (𝐹𝐽) ∈ Q)
89 addclnq 7203 . . . . 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 409 . . . 4 (𝜑 → (((𝐹𝐵) +Q (*Q‘[⟨𝐵, 1o⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1o⟩] ~Q )) ∈ Q)
91 so2nr 4247 . . . . 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 421 . . . 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 409 . . 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 680 . . 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 133 . 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 103   ↔ wb 104   ∨ w3o 962   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ⟨cop 3531   class class class wbr 3933   Or wor 4221  ⟶wf 5123  ‘cfv 5127  (class class class)co 5778  1oc1o 6310  [cec 6431  Ncnpi 7100
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