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Theorem caucvgprprlemnbj 6789
Description: Lemma for caucvgprpr 6808. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 17-Jun-2021.)
Hypotheses
Ref Expression
caucvgprpr.f (𝜑𝐹:NP)
caucvgprpr.cau (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
caucvgprprlemnbj.b (𝜑𝐵N)
caucvgprprlemnbj.j (𝜑𝐽N)
Assertion
Ref Expression
caucvgprprlemnbj (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))
Distinct variable groups:   𝐵,𝑘,𝑙,𝑛   𝑢,𝐵,𝑘,𝑛   𝑘,𝐹,𝑛   𝑘,𝐽,𝑙,𝑛   𝑢,𝐽
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑙)   𝐹(𝑢,𝑙)

Proof of Theorem caucvgprprlemnbj
Dummy variables 𝑝 𝑞 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 caucvgprpr.f . . . . . . 7 (𝜑𝐹:NP)
2 caucvgprpr.cau . . . . . . 7 (𝜑 → ∀𝑛N𝑘N (𝑛 <N 𝑘 → ((𝐹𝑛)<P ((𝐹𝑘) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑘)<P ((𝐹𝑛) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑛, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1𝑜⟩] ~Q ) <Q 𝑢}⟩))))
31, 2caucvgprprlemval 6784 . . . . . 6 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
43simprd 107 . . . . 5 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
5 caucvgprprlemnbj.b . . . . . . . . 9 (𝜑𝐵N)
61, 5ffvelrnd 5303 . . . . . . . 8 (𝜑 → (𝐹𝐵) ∈ P)
7 recnnpr 6644 . . . . . . . . 9 (𝐵N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
85, 7syl 14 . . . . . . . 8 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
9 addclpr 6633 . . . . . . . 8 (((𝐹𝐵) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
106, 8, 9syl2anc 391 . . . . . . 7 (𝜑 → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
11 caucvgprprlemnbj.j . . . . . . . 8 (𝜑𝐽N)
12 recnnpr 6644 . . . . . . . 8 (𝐽N → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
1311, 12syl 14 . . . . . . 7 (𝜑 → ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P)
14 ltaddpr 6693 . . . . . . 7 ((((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
1510, 13, 14syl2anc 391 . . . . . 6 (𝜑 → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
1615adantr 261 . . . . 5 ((𝜑𝐵 <N 𝐽) → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
17 ltsopr 6692 . . . . . 6 <P Or P
18 ltrelpr 6601 . . . . . 6 <P ⊆ (P × P)
1917, 18sotri 4720 . . . . 5 (((𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
204, 16, 19syl2anc 391 . . . 4 ((𝜑𝐵 <N 𝐽) → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
21 ltaddpr 6693 . . . . . . . 8 (((𝐹𝐵) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
226, 8, 21syl2anc 391 . . . . . . 7 (𝜑 → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2322adantr 261 . . . . . 6 ((𝜑𝐵 = 𝐽) → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
24 fveq2 5178 . . . . . . . 8 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
2524breq1d 3774 . . . . . . 7 (𝐵 = 𝐽 → ((𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
2625adantl 262 . . . . . 6 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
2723, 26mpbid 135 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2815adantr 261 . . . . 5 ((𝜑𝐵 = 𝐽) → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2927, 28, 19syl2anc 391 . . . 4 ((𝜑𝐵 = 𝐽) → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
301, 2caucvgprprlemval 6784 . . . . . 6 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐽) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
3130simpld 105 . . . . 5 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
32 ltaprg 6715 . . . . . . . . 9 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
3332adantl 262 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P𝑧P)) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
34 addcomprg 6674 . . . . . . . . 9 ((𝑥P𝑦P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3534adantl 262 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3633, 6, 10, 13, 35caovord2d 5670 . . . . . . 7 (𝜑 → ((𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ↔ ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)))
3722, 36mpbid 135 . . . . . 6 (𝜑 → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
3837adantr 261 . . . . 5 ((𝜑𝐽 <N 𝐵) → ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
3917, 18sotri 4720 . . . . 5 (((𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)) → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
4031, 38, 39syl2anc 391 . . . 4 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
41 pitri3or 6418 . . . . 5 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
425, 11, 41syl2anc 391 . . . 4 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
4320, 29, 40, 42mpjao3dan 1202 . . 3 (𝜑 → (𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
441, 11ffvelrnd 5303 . . . . 5 (𝜑 → (𝐹𝐽) ∈ P)
45 addclpr 6633 . . . . . 6 ((((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P ∧ ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ ∈ P) → (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
4610, 13, 45syl2anc 391 . . . . 5 (𝜑 → (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)
47 so2nr 4058 . . . . . 6 ((<P Or P ∧ ((𝐹𝐽) ∈ P ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P)) → ¬ ((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
4817, 47mpan 400 . . . . 5 (((𝐹𝐽) ∈ P ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∈ P) → ¬ ((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
4944, 46, 48syl2anc 391 . . . 4 (𝜑 → ¬ ((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
50 imnan 624 . . . 4 (((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) → ¬ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)) ↔ ¬ ((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) ∧ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
5149, 50sylibr 137 . . 3 (𝜑 → ((𝐹𝐽)<P (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) → ¬ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽)))
5243, 51mpd 13 . 2 (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽))
53 breq1 3767 . . . . . . 7 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
5453cbvabv 2161 . . . . . 6 {𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}
55 breq2 3768 . . . . . . 7 (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢))
5655cbvabv 2161 . . . . . 6 {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}
5754, 56opeq12i 3554 . . . . 5 ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩
5857oveq2i 5523 . . . 4 ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
59 breq1 3767 . . . . . 6 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6059cbvabv 2161 . . . . 5 {𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}
61 breq2 3768 . . . . . 6 (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢))
6261cbvabv 2161 . . . . 5 {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}
6360, 62opeq12i 3554 . . . 4 ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩
6458, 63oveq12i 5524 . . 3 (((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) = (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)
6564breq1i 3771 . 2 ((((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩)<P (𝐹𝐽) ↔ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))
6652, 65sylnib 601 1 (𝜑 → ¬ (((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}⟩) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}⟩)<P (𝐹𝐽))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  w3o 884  w3a 885   = wceq 1243  wcel 1393  {cab 2026  wral 2306  cop 3378   class class class wbr 3764   Or wor 4032  wf 4898  cfv 4902  (class class class)co 5512  1𝑜c1o 5994  [cec 6104  Ncnpi 6368   <N clti 6371   ~Q ceq 6375  *Qcrq 6380   <Q cltq 6381  Pcnp 6387   +P cpp 6389  <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  caucvgprprlemaddq  6804
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