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Theorem caucvgprprlemnbj 7717
Description: Lemma for caucvgprpr 7736. 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 7712 . . . . . 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 114 . . . . 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, 5ffvelcdmd 5669 . . . . . . . 8 (𝜑 → (𝐹𝐵) ∈ P)
7 recnnpr 7572 . . . . . . . . 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 7561 . . . . . . . 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 411 . . . . . . 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 7572 . . . . . . . 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 7621 . . . . . . 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 411 . . . . . 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 276 . . . . 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 7620 . . . . . 6 <P Or P
18 ltrelpr 7529 . . . . . 6 <P ⊆ (P × P)
1917, 18sotri 5039 . . . . 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 411 . . . 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 7621 . . . . . . . 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 411 . . . . . . 7 (𝜑 → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞}⟩))
2322adantr 276 . . . . . 6 ((𝜑𝐵 = 𝐽) → (𝐹𝐵)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞}⟩))
24 fveq2 5531 . . . . . . . 8 (𝐵 = 𝐽 → (𝐹𝐵) = (𝐹𝐽))
2524breq1d 4028 . . . . . . 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 277 . . . . . 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 147 . . . . 5 ((𝜑𝐵 = 𝐽) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞}⟩))
2815adantr 276 . . . . 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 411 . . . 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 7712 . . . . . 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 112 . . . . 5 ((𝜑𝐽 <N 𝐵) → (𝐹𝐽)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞}⟩))
32 ltaprg 7643 . . . . . . . . 9 ((𝑥P𝑦P𝑧P) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
3332adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P𝑧P)) → (𝑥<P 𝑦 ↔ (𝑧 +P 𝑥)<P (𝑧 +P 𝑦)))
34 addcomprg 7602 . . . . . . . . 9 ((𝑥P𝑦P) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3534adantl 277 . . . . . . . 8 ((𝜑 ∧ (𝑥P𝑦P)) → (𝑥 +P 𝑦) = (𝑦 +P 𝑥))
3633, 6, 10, 13, 35caovord2d 6062 . . . . . . 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 147 . . . . . 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 276 . . . . 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 5039 . . . . 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 411 . . . 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 7346 . . . . 5 ((𝐵N𝐽N) → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
425, 11, 41syl2anc 411 . . . 4 (𝜑 → (𝐵 <N 𝐽𝐵 = 𝐽𝐽 <N 𝐵))
4320, 29, 40, 42mpjao3dan 1318 . . 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, 11ffvelcdmd 5669 . . . . 5 (𝜑 → (𝐹𝐽) ∈ P)
45 addclpr 7561 . . . . . 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 411 . . . . 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 4336 . . . . . 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 424 . . . . 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 411 . . . 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 691 . . . 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 134 . . 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 4021 . . . . . . 7 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )))
5453cbvabv 2314 . . . . . 6 {𝑝𝑝 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐵, 1o⟩] ~Q )}
55 breq2 4022 . . . . . . 7 (𝑞 = 𝑢 → ((*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢))
5655cbvabv 2314 . . . . . 6 {𝑞 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐵, 1o⟩] ~Q ) <Q 𝑢}
5754, 56opeq12i 3798 . . . . 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 5903 . . . 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 4021 . . . . . 6 (𝑝 = 𝑙 → (𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )))
6059cbvabv 2314 . . . . 5 {𝑝𝑝 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐽, 1o⟩] ~Q )}
61 breq2 4022 . . . . . 6 (𝑞 = 𝑢 → ((*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞 ↔ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢))
6261cbvabv 2314 . . . . 5 {𝑞 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑞} = {𝑢 ∣ (*Q‘[⟨𝐽, 1o⟩] ~Q ) <Q 𝑢}
6360, 62opeq12i 3798 . . . 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 5904 . . 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 4025 . 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 677 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 104  wb 105  w3o 979  w3a 980   = wceq 1364  wcel 2160  {cab 2175  wral 2468  cop 3610   class class class wbr 4018   Or wor 4310  wf 5228  cfv 5232  (class class class)co 5892  1oc1o 6429  [cec 6552  Ncnpi 7296   <N clti 7299   ~Q ceq 7303  *Qcrq 7308   <Q cltq 7309  Pcnp 7315   +P cpp 7317  <P cltp 7319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-irdg 6390  df-1o 6436  df-2o 6437  df-oadd 6440  df-omul 6441  df-er 6554  df-ec 6556  df-qs 6560  df-ni 7328  df-pli 7329  df-mi 7330  df-lti 7331  df-plpq 7368  df-mpq 7369  df-enq 7371  df-nqqs 7372  df-plqqs 7373  df-mqqs 7374  df-1nqqs 7375  df-rq 7376  df-ltnqqs 7377  df-enq0 7448  df-nq0 7449  df-0nq0 7450  df-plq0 7451  df-mq0 7452  df-inp 7490  df-iplp 7492  df-iltp 7494
This theorem is referenced by:  caucvgprprlemaddq  7732
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