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Theorem caucvgprprlemnbj 6849
Description: Lemma for caucvgprpr 6868. 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 6844 . . . . . 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 111 . . . . 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 5331 . . . . . . . 8 (𝜑 → (𝐹𝐵) ∈ P)
7 recnnpr 6704 . . . . . . . . 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 6693 . . . . . . . 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 397 . . . . . . 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 6704 . . . . . . . 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 6753 . . . . . . 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 397 . . . . . 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 265 . . . . 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 6752 . . . . . 6 <P Or P
18 ltrelpr 6661 . . . . . 6 <P ⊆ (P × P)
1917, 18sotri 4748 . . . . 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 397 . . . 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 6753 . . . . . . . 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 397 . . . . . . 7 (𝜑 → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2322adantr 265 . . . . . 6 ((𝜑𝐵 = 𝐽) → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
24 fveq2 5206 . . . . . . . 8 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
2524breq1d 3802 . . . . . . 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 266 . . . . . 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 139 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
2815adantr 265 . . . . 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 397 . . . 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 6844 . . . . . 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 109 . . . . 5 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞}⟩))
32 ltaprg 6775 . . . . . . . . 9 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
3332adantl 266 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P𝑧P)) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
34 addcomprg 6734 . . . . . . . . 9 ((𝑥P𝑦P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3534adantl 266 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3633, 6, 10, 13, 35caovord2d 5698 . . . . . . 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 139 . . . . . 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 265 . . . . 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 4748 . . . . 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 397 . . . 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 6478 . . . . 5 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
425, 11, 41syl2anc 397 . . . 4 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
4320, 29, 40, 42mpjao3dan 1213 . . 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 5331 . . . . 5 (𝜑 → (𝐹𝐽) ∈ P)
45 addclpr 6693 . . . . . 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 397 . . . . 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 4086 . . . . . 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 408 . . . . 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 397 . . . 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 634 . . . 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 141 . . 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 3795 . . . . . . 7 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )))
5453cbvabv 2177 . . . . . 6 {𝑝𝑝 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐵, 1𝑜⟩] ~Q )}
55 breq2 3796 . . . . . . 7 (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢))
5655cbvabv 2177 . . . . . 6 {𝑞 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1𝑜⟩] ~Q ) <Q 𝑢}
5754, 56opeq12i 3582 . . . . 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 5551 . . . 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 3795 . . . . . 6 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )))
6059cbvabv 2177 . . . . 5 {𝑝𝑝 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐽, 1𝑜⟩] ~Q )}
61 breq2 3796 . . . . . 6 (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢))
6261cbvabv 2177 . . . . 5 {𝑞 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1𝑜⟩] ~Q ) <Q 𝑢}
6360, 62opeq12i 3582 . . . 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 5552 . . 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 3799 . 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 611 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 101  wb 102  w3o 895  w3a 896   = wceq 1259  wcel 1409  {cab 2042  wral 2323  cop 3406   class class class wbr 3792   Or wor 4060  wf 4926  cfv 4930  (class class class)co 5540  1𝑜c1o 6025  [cec 6135  Ncnpi 6428   <N clti 6431   ~Q ceq 6435  *Qcrq 6440   <Q cltq 6441  Pcnp 6447   +P cpp 6449  <P cltp 6451
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 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
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 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  caucvgprprlemaddq  6864
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