Theorem caucvgprprlemval 7755

Description: Lemma for caucvgprpr 7779. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	⊢ (𝜑 → 𝐹:N⟶P)
caucvgprpr.cau	⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))

Assertion

Ref	Expression
caucvgprprlemval	⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <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.

Step	Hyp	Ref	Expression
1		ltrelpi 7391	. . . . 5 ⊢ <N ⊆ (N × N)
2	1	brel 4715	. . . 4 ⊢ (𝐴 <N 𝐵 → (𝐴 ∈ N ∧ 𝐵 ∈ N))
3	2	adantl 277	. . 3 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → (𝐴 ∈ N ∧ 𝐵 ∈ N))
4		caucvgprpr.f	. . . . 5 ⊢ (𝜑 → 𝐹:N⟶P)
5		caucvgprpr.cau	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ N ∀𝑘 ∈ N (𝑛 <N 𝑘 → ((𝐹‘𝑛)<P ((𝐹‘𝑘) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑘)<P ((𝐹‘𝑛) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑛, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑛, 1o⟩] ~Q ) <Q 𝑢}⟩))))
6	4, 5	caucvgprprlemcbv 7754	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
7	6	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → ∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))))
8		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → 𝐴 <N 𝐵)
9		breq1 4036	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 <N 𝑏 ↔ 𝐴 <N 𝑏))
10		fveq2 5558	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
11		opeq1 3808	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 1o⟩ = ⟨𝐴, 1o⟩)
12	11	eceq1d 6628	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → [⟨𝑎, 1o⟩] ~Q = [⟨𝐴, 1o⟩] ~Q )
13	12	fveq2d 5562	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝐴, 1o⟩] ~Q ))
14	13	breq2d 4045	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
15	14	abbidv 2314	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )})
16	13	breq1d 4043	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢))
17	16	abbidv 2314	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢})
18	15, 17	opeq12d 3816	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)
19	18	oveq2d 5938	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
20	10, 19	breq12d 4046	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
21	10, 18	oveq12d 5940	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
22	21	breq2d 4045	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
23	20, 22	anbi12d 473	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))))
24	9, 23	imbi12d 234	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝐴 <N 𝑏 → ((𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
25		breq2 4037	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <N 𝑏 ↔ 𝐴 <N 𝐵))
26		fveq2 5558	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
27	26	oveq1d 5937	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))
28	27	breq2d 4045	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
29	26	breq1d 4043	. . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
30	28, 29	anbi12d 473	. . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))))
31	25, 30	imbi12d 234	. . . 4 ⊢ (𝑏 = 𝐵 → ((𝐴 <N 𝑏 → ((𝐹‘𝐴)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩))) ↔ (𝐴 <N 𝐵 → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
32	24, 31	rspc2v 2881	. . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (∀𝑎 ∈ N ∀𝑏 ∈ N (𝑎 <N 𝑏 → ((𝐹‘𝑎)<P ((𝐹‘𝑏) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝑏)<P ((𝐹‘𝑎) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢}⟩))) → (𝐴 <N 𝐵 → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))))
33	3, 7, 8, 32	syl3c 63	. 2 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)))
34		breq1 4036	. . . . . . 7 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
35	34	cbvabv 2321	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}
36		breq2 4037	. . . . . . 7 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞))
37	36	cbvabv 2321	. . . . . 6 ⊢ {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}
38	35, 37	opeq12i 3813	. . . . 5 ⊢ ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩ = ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩
39	38	oveq2i 5933	. . . 4 ⊢ ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)
40	39	breq2i 4041	. . 3 ⊢ ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩))
41	38	oveq2i 5933	. . . 4 ⊢ ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) = ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)
42	41	breq2i 4041	. . 3 ⊢ ((𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ↔ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩))
43	40, 42	anbi12i 460	. 2 ⊢ (((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢}⟩)) ↔ ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))
44	33, 43	sylib 122	1 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → ((𝐹‘𝐴)<P ((𝐹‘𝐵) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩) ∧ (𝐹‘𝐵)<P ((𝐹‘𝐴) +P ⟨{𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}, {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}⟩)))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  {cab 2182  ∀wral 2475  ⟨cop 3625   class class class wbr 4033  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1_oc1o 6467  [cec 6590  Ncnpi 7339   <_N clti 7342   ~_Q ceq 7346  *_Qcrq 7351   <_Q cltq 7352  Pcnp 7358   +_P cpp 7360  <_P cltp 7362

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-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fv 5266  df-ov 5925  df-ec 6594  df-lti 7374

This theorem is referenced by:  caucvgprprlemnkltj  7756  caucvgprprlemnjltk  7758  caucvgprprlemnbj  7760