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Theorem caucvgprprlemval 7508
Description: Lemma for caucvgprpr 7532. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-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}⟩))))
Assertion
Ref Expression
caucvgprprlemval ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))
Distinct variable groups:   𝐴,𝑙   𝑢,𝐴   𝐴,𝑝,𝑙   𝐴,𝑞,𝑢   𝑘,𝐹,𝑛   𝑘,𝑙,𝑛   𝑢,𝑘,𝑛
Allowed substitution hints:   𝜑(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐴(𝑘,𝑛)   𝐵(𝑢,𝑘,𝑛,𝑞,𝑝,𝑙)   𝐹(𝑢,𝑞,𝑝,𝑙)

Proof of Theorem caucvgprprlemval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpi 7144 . . . . 5 <N ⊆ (N × N)
21brel 4591 . . . 4 (𝐴 <N 𝐵 → (𝐴N𝐵N))
32adantl 275 . . 3 ((𝜑𝐴 <N 𝐵) → (𝐴N𝐵N))
4 caucvgprpr.f . . . . 5 (𝜑𝐹:NP)
5 caucvgprpr.cau . . . . 5 (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}⟩))))
64, 5caucvgprprlemcbv 7507 . . . 4 (𝜑 → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
76adantr 274 . . 3 ((𝜑𝐴 <N 𝐵) → ∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
8 simpr 109 . . 3 ((𝜑𝐴 <N 𝐵) → 𝐴 <N 𝐵)
9 breq1 3932 . . . . 5 (𝑎 = 𝐴 → (𝑎 <N 𝑏𝐴 <N 𝑏))
10 fveq2 5421 . . . . . . 7 (𝑎 = 𝐴 → (𝐹𝑎) = (𝐹𝐴))
11 opeq1 3705 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ⟨𝑎, 1o⟩ = ⟨𝐴, 1o⟩)
1211eceq1d 6465 . . . . . . . . . . . 12 (𝑎 = 𝐴 → [⟨𝑎, 1o⟩] ~Q = [⟨𝐴, 1o⟩] ~Q )
1312fveq2d 5425 . . . . . . . . . . 11 (𝑎 = 𝐴 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝐴, 1o⟩] ~Q ))
1413breq2d 3941 . . . . . . . . . 10 (𝑎 = 𝐴 → (𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
1514abbidv 2257 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )})
1613breq1d 3939 . . . . . . . . . 10 (𝑎 = 𝐴 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢))
1716abbidv 2257 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢})
1815, 17opeq12d 3713 . . . . . . . 8 (𝑎 = 𝐴 → ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)
1918oveq2d 5790 . . . . . . 7 (𝑎 = 𝐴 → ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
2010, 19breq12d 3942 . . . . . 6 (𝑎 = 𝐴 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
2110, 18oveq12d 5792 . . . . . . 7 (𝑎 = 𝐴 → ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
2221breq2d 3941 . . . . . 6 (𝑎 = 𝐴 → ((𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
2320, 22anbi12d 464 . . . . 5 (𝑎 = 𝐴 → (((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))))
249, 23imbi12d 233 . . . 4 (𝑎 = 𝐴 → ((𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝐴 <N 𝑏 → ((𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
25 breq2 3933 . . . . 5 (𝑏 = 𝐵 → (𝐴 <N 𝑏𝐴 <N 𝐵))
26 fveq2 5421 . . . . . . . 8 (𝑏 = 𝐵 → (𝐹𝑏) = (𝐹𝐵))
2726oveq1d 5789 . . . . . . 7 (𝑏 = 𝐵 → ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
2827breq2d 3941 . . . . . 6 (𝑏 = 𝐵 → ((𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
2926breq1d 3939 . . . . . 6 (𝑏 = 𝐵 → ((𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
3028, 29anbi12d 464 . . . . 5 (𝑏 = 𝐵 → (((𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))))
3125, 30imbi12d 233 . . . 4 (𝑏 = 𝐵 → ((𝐴 <N 𝑏 → ((𝐹𝐴)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝐴 <N 𝐵 → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
3224, 31rspc2v 2802 . . 3 ((𝐴N𝐵N) → (∀𝑎N𝑏N (𝑎 <N 𝑏 → ((𝐹𝑎)<P ((𝐹𝑏) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝑏)<P ((𝐹𝑎) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))) → (𝐴 <N 𝐵 → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
333, 7, 8, 32syl3c 63 . 2 ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
34 breq1 3932 . . . . . . 7 (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
3534cbvabv 2264 . . . . . 6 {𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )} = {𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}
36 breq2 3933 . . . . . . 7 (𝑢 = 𝑞 → ((*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞))
3736cbvabv 2264 . . . . . 6 {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}
3835, 37opeq12i 3710 . . . . 5 ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩
3938oveq2i 5785 . . . 4 ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)
4039breq2i 3937 . . 3 ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩))
4138oveq2i 5785 . . . 4 ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)
4241breq2i 3937 . . 3 ((𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩))
4340, 42anbi12i 455 . 2 (((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑙𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))
4433, 43sylib 121 1 ((𝜑𝐴 <N 𝐵) → ((𝐹𝐴)<P ((𝐹𝐵) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹𝐵)<P ((𝐹𝐴) +P ⟨{𝑝𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {cab 2125  wral 2416  cop 3530   class class class wbr 3929  wf 5119  cfv 5123  (class class class)co 5774  1oc1o 6306  [cec 6427  Ncnpi 7092   <N clti 7095   ~Q ceq 7099  *Qcrq 7104   <Q cltq 7105  Pcnp 7111   +P cpp 7113  <P cltp 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fv 5131  df-ov 5777  df-ec 6431  df-lti 7127
This theorem is referenced by:  caucvgprprlemnkltj  7509  caucvgprprlemnjltk  7511  caucvgprprlemnbj  7513
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