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Theorem caucvgprlemnkj 6821
Description: Lemma for caucvgpr 6837. Part of disjointness. (Contributed by Jim Kingdon, 23-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 )))))
caucvgprlemnkj.k (𝜑𝐾N)
caucvgprlemnkj.j (𝜑𝐽N)
caucvgprlemnkj.s (𝜑𝑆Q)
Assertion
Ref Expression
caucvgprlemnkj (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +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 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltsonq 6553 . . . 4 <Q Or Q
2 ltrelnq 6520 . . . 4 <Q ⊆ (Q × Q)
31, 2son2lpi 4748 . . 3 ¬ (𝑆 <Q (𝐹𝐽) ∧ (𝐹𝐽) <Q 𝑆)
4 simprl 491 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾))
5 caucvgpr.cau . . . . . . . . . . . 12 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))))
6 breq1 3794 . . . . . . . . . . . . . 14 (𝑛 = 𝑎 → (𝑛 <N 𝑘𝑎 <N 𝑘))
7 fveq2 5205 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → (𝐹𝑛) = (𝐹𝑎))
8 opeq1 3576 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑎 → ⟨𝑛, 1𝑜⟩ = ⟨𝑎, 1𝑜⟩)
98eceq1d 6172 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑎 → [⟨𝑛, 1𝑜⟩] ~Q = [⟨𝑎, 1𝑜⟩] ~Q )
109fveq2d 5209 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑎 → (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))
1110oveq2d 5555 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
127, 11breq12d 3804 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
137, 10oveq12d 5557 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑎 → ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) = ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
1413breq2d 3803 . . . . . . . . . . . . . . 15 (𝑛 = 𝑎 → ((𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ↔ (𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
1512, 14anbi12d 450 . . . . . . . . . . . . . 14 (𝑛 = 𝑎 → (((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
166, 15imbi12d 227 . . . . . . . . . . . . 13 (𝑛 = 𝑎 → ((𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ (𝑎 <N 𝑘 → ((𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))))
17 breq2 3795 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → (𝑎 <N 𝑘𝑎 <N 𝑏))
18 fveq2 5205 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑏 → (𝐹𝑘) = (𝐹𝑏))
1918oveq1d 5554 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑏 → ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))
2019breq2d 3803 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
2118breq1d 3801 . . . . . . . . . . . . . . 15 (𝑘 = 𝑏 → ((𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))
2220, 21anbi12d 450 . . . . . . . . . . . . . 14 (𝑘 = 𝑏 → (((𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
2317, 22imbi12d 227 . . . . . . . . . . . . 13 (𝑘 = 𝑏 → ((𝑎 <N 𝑘 → ((𝐹𝑎) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) ↔ (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))))))
2416, 23cbvral2v 2558 . . . . . . . . . . . 12 (∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛) <Q ((𝐹𝑘) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )) ∧ (𝐹𝑘) <Q ((𝐹𝑛) +Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )))) ↔ ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
255, 24sylib 131 . . . . . . . . . . 11 (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))))
26 caucvgprlemnkj.k . . . . . . . . . . . 12 (𝜑𝐾N)
27 caucvgprlemnkj.j . . . . . . . . . . . 12 (𝜑𝐽N)
28 breq1 3794 . . . . . . . . . . . . . 14 (𝑎 = 𝐾 → (𝑎 <N 𝑏𝐾 <N 𝑏))
29 fveq2 5205 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐾 → (𝐹𝑎) = (𝐹𝐾))
30 opeq1 3576 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐾 → ⟨𝑎, 1𝑜⟩ = ⟨𝐾, 1𝑜⟩)
3130eceq1d 6172 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝐾 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝐾, 1𝑜⟩] ~Q )
3231fveq2d 5209 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐾 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))
3332oveq2d 5555 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐾 → ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
3429, 33breq12d 3804 . . . . . . . . . . . . . . 15 (𝑎 = 𝐾 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
3529, 32oveq12d 5557 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐾 → ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
3635breq2d 3803 . . . . . . . . . . . . . . 15 (𝑎 = 𝐾 → ((𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
3734, 36anbi12d 450 . . . . . . . . . . . . . 14 (𝑎 = 𝐾 → (((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
3828, 37imbi12d 227 . . . . . . . . . . . . 13 (𝑎 = 𝐾 → ((𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) ↔ (𝐾 <N 𝑏 → ((𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
39 breq2 3795 . . . . . . . . . . . . . 14 (𝑏 = 𝐽 → (𝐾 <N 𝑏𝐾 <N 𝐽))
40 fveq2 5205 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝐽 → (𝐹𝑏) = (𝐹𝐽))
4140oveq1d 5554 . . . . . . . . . . . . . . . 16 (𝑏 = 𝐽 → ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
4241breq2d 3803 . . . . . . . . . . . . . . 15 (𝑏 = 𝐽 → ((𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
4340breq1d 3801 . . . . . . . . . . . . . . 15 (𝑏 = 𝐽 → ((𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
4442, 43anbi12d 450 . . . . . . . . . . . . . 14 (𝑏 = 𝐽 → (((𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
4539, 44imbi12d 227 . . . . . . . . . . . . 13 (𝑏 = 𝐽 → ((𝐾 <N 𝑏 → ((𝐹𝐾) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))) ↔ (𝐾 <N 𝐽 → ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
4638, 45rspc2v 2684 . . . . . . . . . . . 12 ((𝐾N𝐽N) → (∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐾 <N 𝐽 → ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
4726, 27, 46syl2anc 397 . . . . . . . . . . 11 (𝜑 → (∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐾 <N 𝐽 → ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))))
4825, 47mpd 13 . . . . . . . . . 10 (𝜑 → (𝐾 <N 𝐽 → ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))))
4948imp 119 . . . . . . . . 9 ((𝜑𝐾 <N 𝐽) → ((𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
5049simpld 109 . . . . . . . 8 ((𝜑𝐾 <N 𝐽) → (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
5150adantr 265 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
521, 2sotri 4747 . . . . . . 7 (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
534, 51, 52syl2anc 397 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
54 ltanqg 6555 . . . . . . . 8 ((𝑓Q𝑔QQ) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
5554adantl 266 . . . . . . 7 ((((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
56 caucvgprlemnkj.s . . . . . . . 8 (𝜑𝑆Q)
5756ad2antrr 465 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆Q)
58 caucvgpr.f . . . . . . . . 9 (𝜑𝐹:NQ)
5958, 27ffvelrnd 5330 . . . . . . . 8 (𝜑 → (𝐹𝐽) ∈ Q)
6059ad2antrr 465 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹𝐽) ∈ Q)
61 nnnq 6577 . . . . . . . . 9 (𝐾N → [⟨𝐾, 1𝑜⟩] ~QQ)
62 recclnq 6547 . . . . . . . . 9 ([⟨𝐾, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
6326, 61, 623syl 17 . . . . . . . 8 (𝜑 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
6463ad2antrr 465 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
65 addcomnqg 6536 . . . . . . . 8 ((𝑓Q𝑔Q) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
6665adantl 266 . . . . . . 7 ((((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
6755, 57, 60, 64, 66caovord2d 5697 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q (𝐹𝐽) ↔ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ))))
6853, 67mpbird 160 . . . . 5 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q (𝐹𝐽))
69 nnnq 6577 . . . . . . . . 9 (𝐽N → [⟨𝐽, 1𝑜⟩] ~QQ)
70 recclnq 6547 . . . . . . . . 9 ([⟨𝐽, 1𝑜⟩] ~QQ → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
7127, 69, 703syl 17 . . . . . . . 8 (𝜑 → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
7271ad2antrr 465 . . . . . . 7 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q)
73 ltaddnq 6562 . . . . . . 7 (((𝐹𝐽) ∈ Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (𝐹𝐽) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
7460, 72, 73syl2anc 397 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹𝐽) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
75 simprr 492 . . . . . 6 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
761, 2sotri 4747 . . . . . 6 (((𝐹𝐽) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝐹𝐽) <Q 𝑆)
7774, 75, 76syl2anc 397 . . . . 5 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹𝐽) <Q 𝑆)
7868, 77jca 294 . . . 4 (((𝜑𝐾 <N 𝐽) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q (𝐹𝐽) ∧ (𝐹𝐽) <Q 𝑆))
7978ex 112 . . 3 ((𝜑𝐾 <N 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝑆 <Q (𝐹𝐽) ∧ (𝐹𝐽) <Q 𝑆)))
803, 79mtoi 600 . 2 ((𝜑𝐾 <N 𝐽) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
811, 2son2lpi 4748 . . 3 ¬ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
82 opeq1 3576 . . . . . . . . . . . 12 (𝐾 = 𝐽 → ⟨𝐾, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
8382eceq1d 6172 . . . . . . . . . . 11 (𝐾 = 𝐽 → [⟨𝐾, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
8483fveq2d 5209 . . . . . . . . . 10 (𝐾 = 𝐽 → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
8584oveq2d 5555 . . . . . . . . 9 (𝐾 = 𝐽 → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) = (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
86 fveq2 5205 . . . . . . . . 9 (𝐾 = 𝐽 → (𝐹𝐾) = (𝐹𝐽))
8785, 86breq12d 3804 . . . . . . . 8 (𝐾 = 𝐽 → ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ↔ (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽)))
8887anbi1d 446 . . . . . . 7 (𝐾 = 𝐽 → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
8988adantl 266 . . . . . 6 ((𝜑𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
9054adantl 266 . . . . . . . . 9 ((𝜑 ∧ (𝑓Q𝑔QQ)) → (𝑓 <Q 𝑔 ↔ ( +Q 𝑓) <Q ( +Q 𝑔)))
91 addclnq 6530 . . . . . . . . . 10 ((𝑆Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
9256, 71, 91syl2anc 397 . . . . . . . . 9 (𝜑 → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∈ Q)
9365adantl 266 . . . . . . . . 9 ((𝜑 ∧ (𝑓Q𝑔Q)) → (𝑓 +Q 𝑔) = (𝑔 +Q 𝑓))
9490, 92, 59, 71, 93caovord2d 5697 . . . . . . . 8 (𝜑 → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
9594adantr 265 . . . . . . 7 ((𝜑𝐾 = 𝐽) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽) ↔ ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
9695anbi1d 446 . . . . . 6 ((𝜑𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q (𝐹𝐽) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
9789, 96bitrd 181 . . . . 5 ((𝜑𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) ↔ (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
981, 2sotri 4747 . . . . 5 ((((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
9997, 98syl6bi 156 . . . 4 ((𝜑𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
100 ltaddnq 6562 . . . . . . 7 ((𝑆Q ∧ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ∈ Q) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
10156, 71, 100syl2anc 397 . . . . . 6 (𝜑𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
102 ltaddnq 6562 . . . . . . 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 )))
10392, 71, 102syl2anc 397 . . . . . 6 (𝜑 → (𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
1041, 2sotri 4747 . . . . . 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 )))
105101, 103, 104syl2anc 397 . . . . 5 (𝜑𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
106105adantr 265 . . . 4 ((𝜑𝐾 = 𝐽) → 𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
10799, 106jctird 304 . . 3 ((𝜑𝐾 = 𝐽) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆𝑆 <Q ((𝑆 +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
10881, 107mtoi 600 . 2 ((𝜑𝐾 = 𝐽) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
1091, 2son2lpi 4748 . . 3 ¬ (𝑆 <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
11056ad2antrr 465 . . . . . . 7 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆Q)
11163ad2antrr 465 . . . . . . 7 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q)
112 ltaddnq 6562 . . . . . . 7 ((𝑆Q ∧ (*Q‘[⟨𝐾, 1𝑜⟩] ~Q ) ∈ Q) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
113110, 111, 112syl2anc 397 . . . . . 6 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )))
114 simprl 491 . . . . . . 7 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾))
115 breq1 3794 . . . . . . . . . . . . . 14 (𝑎 = 𝐽 → (𝑎 <N 𝑏𝐽 <N 𝑏))
116 fveq2 5205 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐽 → (𝐹𝑎) = (𝐹𝐽))
117 opeq1 3576 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐽 → ⟨𝑎, 1𝑜⟩ = ⟨𝐽, 1𝑜⟩)
118117eceq1d 6172 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝐽 → [⟨𝑎, 1𝑜⟩] ~Q = [⟨𝐽, 1𝑜⟩] ~Q )
119118fveq2d 5209 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝐽 → (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ) = (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))
120119oveq2d 5555 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐽 → ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
121116, 120breq12d 3804 . . . . . . . . . . . . . . 15 (𝑎 = 𝐽 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
122116, 119oveq12d 5557 . . . . . . . . . . . . . . . 16 (𝑎 = 𝐽 → ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) = ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
123122breq2d 3803 . . . . . . . . . . . . . . 15 (𝑎 = 𝐽 → ((𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ↔ (𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
124121, 123anbi12d 450 . . . . . . . . . . . . . 14 (𝑎 = 𝐽 → (((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
125115, 124imbi12d 227 . . . . . . . . . . . . 13 (𝑎 = 𝐽 → ((𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) ↔ (𝐽 <N 𝑏 → ((𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
126 breq2 3795 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (𝐽 <N 𝑏𝐽 <N 𝐾))
127 fveq2 5205 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝐾 → (𝐹𝑏) = (𝐹𝐾))
128127oveq1d 5554 . . . . . . . . . . . . . . . 16 (𝑏 = 𝐾 → ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) = ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
129128breq2d 3803 . . . . . . . . . . . . . . 15 (𝑏 = 𝐾 → ((𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
130127breq1d 3801 . . . . . . . . . . . . . . 15 (𝑏 = 𝐾 → ((𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ↔ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
131129, 130anbi12d 450 . . . . . . . . . . . . . 14 (𝑏 = 𝐾 → (((𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) ↔ ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
132126, 131imbi12d 227 . . . . . . . . . . . . 13 (𝑏 = 𝐾 → ((𝐽 <N 𝑏 → ((𝐹𝐽) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))) ↔ (𝐽 <N 𝐾 → ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
133125, 132rspc2v 2684 . . . . . . . . . . . 12 ((𝐽N𝐾N) → (∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐽 <N 𝐾 → ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
13427, 26, 133syl2anc 397 . . . . . . . . . . 11 (𝜑 → (∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎) <Q ((𝐹𝑏) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )) ∧ (𝐹𝑏) <Q ((𝐹𝑎) +Q (*Q‘[⟨𝑎, 1𝑜⟩] ~Q )))) → (𝐽 <N 𝐾 → ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))))
13525, 134mpd 13 . . . . . . . . . 10 (𝜑 → (𝐽 <N 𝐾 → ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))))
136135imp 119 . . . . . . . . 9 ((𝜑𝐽 <N 𝐾) → ((𝐹𝐽) <Q ((𝐹𝐾) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))))
137136simprd 111 . . . . . . . 8 ((𝜑𝐽 <N 𝐾) → (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
138137adantr 265 . . . . . . 7 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
1391, 2sotri 4747 . . . . . . 7 (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ (𝐹𝐾) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
140114, 138, 139syl2anc 397 . . . . . 6 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
1411, 2sotri 4747 . . . . . 6 ((𝑆 <Q (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) ∧ (𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ))) → 𝑆 <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
142113, 140, 141syl2anc 397 . . . . 5 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → 𝑆 <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
143 simprr 492 . . . . 5 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)
144142, 143jca 294 . . . 4 (((𝜑𝐽 <N 𝐾) ∧ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)) → (𝑆 <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
145144ex 112 . . 3 ((𝜑𝐽 <N 𝐾) → (((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆) → (𝑆 <Q ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆)))
146109, 145mtoi 600 . 2 ((𝜑𝐽 <N 𝐾) → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
147 pitri3or 6477 . . 3 ((𝐾N𝐽N) → (𝐾 <N 𝐽𝐾 = 𝐽𝐽 <N 𝐾))
14826, 27, 147syl2anc 397 . 2 (𝜑 → (𝐾 <N 𝐽𝐾 = 𝐽𝐽 <N 𝐾))
14980, 108, 146, 148mpjao3dan 1213 1 (𝜑 → ¬ ((𝑆 +Q (*Q‘[⟨𝐾, 1𝑜⟩] ~Q )) <Q (𝐹𝐾) ∧ ((𝐹𝐽) +Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )) <Q 𝑆))
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  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:  caucvgprlemdisj  6829
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