Theorem caucvgprprlemval 7489

Description: Lemma for caucvgprpr 7513. 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 7125	. . . . 5 ⊢ <N ⊆ (N × N)
2	1	brel 4586	. . . 4 ⊢ (𝐴 <N 𝐵 → (𝐴 ∈ N ∧ 𝐵 ∈ N))
3	2	adantl 275	. . 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 7488	. . . 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 274	. . 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 109	. . 3 ⊢ ((𝜑 ∧ 𝐴 <N 𝐵) → 𝐴 <N 𝐵)
9		breq1 3927	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑎 <N 𝑏 ↔ 𝐴 <N 𝑏))
10		fveq2 5414	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
11		opeq1 3700	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 1o⟩ = ⟨𝐴, 1o⟩)
12	11	eceq1d 6458	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → [⟨𝑎, 1o⟩] ~Q = [⟨𝐴, 1o⟩] ~Q )
13	12	fveq2d 5418	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (*Q‘[⟨𝑎, 1o⟩] ~Q ) = (*Q‘[⟨𝐴, 1o⟩] ~Q ))
14	13	breq2d 3936	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q ) ↔ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
15	14	abbidv 2255	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝑎, 1o⟩] ~Q )} = {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )})
16	13	breq1d 3934	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢))
17	16	abbidv 2255	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑢 ∣ (*Q‘[⟨𝑎, 1o⟩] ~Q ) <Q 𝑢} = {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢})
18	15, 17	opeq12d 3708	. . . . . . . 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 5783	. . . . . . 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 3937	. . . . . 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 5785	. . . . . . 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 3936	. . . . . 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 464	. . . . 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 233	. . . 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 3928	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝐴 <N 𝑏 ↔ 𝐴 <N 𝐵))
26		fveq2 5414	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐹‘𝑏) = (𝐹‘𝐵))
27	26	oveq1d 5782	. . . . . . 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 3936	. . . . . 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 3934	. . . . . 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 464	. . . . 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 233	. . . 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 2797	. . 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 3927	. . . . . . 7 ⊢ (𝑙 = 𝑝 → (𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q ) ↔ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )))
35	34	cbvabv 2262	. . . . . 6 ⊢ {𝑙 ∣ 𝑙 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )} = {𝑝 ∣ 𝑝 <Q (*Q‘[⟨𝐴, 1o⟩] ~Q )}
36		breq2 3928	. . . . . . 7 ⊢ (𝑢 = 𝑞 → ((*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢 ↔ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞))
37	36	cbvabv 2262	. . . . . 6 ⊢ {𝑢 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑢} = {𝑞 ∣ (*Q‘[⟨𝐴, 1o⟩] ~Q ) <Q 𝑞}
38	35, 37	opeq12i 3705	. . . . 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 5778	. . . 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 3932	. . 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 5778	. . . 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 3932	. . 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 455	. 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 121	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 103   = wceq 1331   ∈ wcel 1480  {cab 2123  ∀wral 2414  ⟨cop 3525   class class class wbr 3924  ⟶wf 5114  ‘cfv 5118  (class class class)co 5767  1_oc1o 6299  [cec 6420  Ncnpi 7073   <_N clti 7076   ~_Q ceq 7080  *_Qcrq 7085   <_Q cltq 7086  Pcnp 7092   +_P cpp 7094  <_P cltp 7096

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 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fv 5126  df-ov 5770  df-ec 6424  df-lti 7108

This theorem is referenced by:  caucvgprprlemnkltj  7490  caucvgprprlemnjltk  7492  caucvgprprlemnbj  7494