Theorem axcaucvglemval 8271

Description: Lemma for axcaucvg 8274. Value of sequence when mapping to N and R. (Contributed by Jim Kingdon, 10-Jul-2021.)

Hypotheses

Ref	Expression
axcaucvg.n	⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
axcaucvg.f	⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
axcaucvg.cau	⊢ (𝜑 → ∀𝑛 ∈ 𝑁 ∀𝑘 ∈ 𝑁 (𝑛 <ℝ 𝑘 → ((𝐹‘𝑛) <ℝ ((𝐹‘𝑘) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)) ∧ (𝐹‘𝑘) <ℝ ((𝐹‘𝑛) + (℩𝑟 ∈ ℝ (𝑛 · 𝑟) = 1)))))
axcaucvg.g	⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))

Assertion

Ref	Expression
axcaucvglemval	⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩)

Distinct variable groups:   𝑗,𝐹,𝑧   𝑧,𝐺   𝑗,𝐽,𝑙,𝑢,𝑧   𝜑,𝑗   𝑦,𝑙,𝑢   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑢, 𝑘, 𝑛, 𝑟, 𝑙)   𝐹(𝑥, 𝑦, 𝑢, 𝑘, 𝑛, 𝑟, 𝑙)   𝐺(𝑥, 𝑦, 𝑢, 𝑗, 𝑘, 𝑛, 𝑟, 𝑙)   𝐽(𝑥, 𝑦, 𝑘, 𝑛, 𝑟)   𝑁(𝑥, 𝑦, 𝑧, 𝑢, 𝑗, 𝑘, 𝑛, 𝑟, 𝑙)

Proof of Theorem axcaucvglemval

Step	Hyp	Ref	Expression
1		axcaucvg.g	. . . . 5 ⊢ 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
2	1	a1i 9	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐺 = (𝑗 ∈ N ↦ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)))
3		opeq1 3904	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ⟨𝑗, 1o⟩ = ⟨𝐽, 1o⟩)
4	3	eceq1d 6847	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
5	4	breq2d 4143	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝑙 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q ))
6	5	abbidv 2358	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q })
7	4	breq1d 4141	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝐽, 1o⟩] ~Q <Q 𝑢))
8	7	abbidv 2358	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢})
9	6, 8	opeq12d 3912	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ = ⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩)
10	9	oveq1d 6103	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
11	10	opeq1d 3910	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩ = ⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩)
12	11	eceq1d 6847	. . . . . . . . 9 ⊢ (𝑗 = 𝐽 → [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R = [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
13	12	opeq1d 3910	. . . . . . . 8 ⊢ (𝑗 = 𝐽 → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
14	13	fveq2d 5702	. . . . . . 7 ⊢ (𝑗 = 𝐽 → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
15	14	eqeq1d 2247	. . . . . 6 ⊢ (𝑗 = 𝐽 → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
16	15	riotabidv 6043	. . . . 5 ⊢ (𝑗 = 𝐽 → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
17	16	adantl 277	. . . 4 ⊢ (((𝜑 ∧ 𝐽 ∈ N) ∧ 𝑗 = 𝐽) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
18		simpr 110	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐽 ∈ N)
19		axcaucvg.n	. . . . 5 ⊢ 𝑁 = ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}
20		axcaucvg.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑁⟶ℝ)
21	19, 20	axcaucvglemcl 8269	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) ∈ R)
22	2, 17, 18, 21	fvmptd 5789	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐺‘𝐽) = (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩))
23	22	eqcomd 2244	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺‘𝐽))
24	22, 21	eqeltrd 2315	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐺‘𝐽) ∈ R)
25	20	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐹:𝑁⟶ℝ)
26		pitonn 8222	. . . . . . . 8 ⊢ (𝐽 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
27	26, 19	eleqtrrdi 2332	. . . . . . 7 ⊢ (𝐽 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
28	27	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ 𝑁)
29	25, 28	ffvelcdmd 5847	. . . . 5 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ)
30		elrealeu 8203	. . . . 5 ⊢ ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ∈ ℝ ↔ ∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
31	29, 30	sylib 122	. . . 4 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
32		eqcom 2240	. . . . 5 ⊢ (⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
33	32	reubii 2739	. . . 4 ⊢ (∃!𝑧 ∈ R ⟨𝑧, 0R⟩ = (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) ↔ ∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
34	31, 33	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩)
35		opeq1 3904	. . . . 5 ⊢ (𝑧 = (𝐺‘𝐽) → ⟨𝑧, 0R⟩ = ⟨(𝐺‘𝐽), 0R⟩)
36	35	eqeq2d 2250	. . . 4 ⊢ (𝑧 = (𝐺‘𝐽) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩ ↔ (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩))
37	36	riota2 6065	. . 3 ⊢ (((𝐺‘𝐽) ∈ R ∧ ∃!𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩ ↔ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺‘𝐽)))
38	24, 34, 37	syl2anc 415	. 2 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → ((𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩ ↔ (℩𝑧 ∈ R (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨𝑧, 0R⟩) = (𝐺‘𝐽)))
39	23, 38	mpbird 167	1 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐹‘⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) = ⟨(𝐺‘𝐽), 0R⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃!wreu 2530  ⟨cop 3712  ∩ cint 3970   class class class wbr 4131   ↦ cmpt 4193  ⟶wf 5376  ‘cfv 5380  ℩crio 6040  (class class class)co 6088  1_oc1o 6684  [cec 6809  Ncnpi 7646   ~_Q ceq 7653   <_Q cltq 7659  1_Pc1p 7666   +_P cpp 7667   ~_R cer 7670  Rcnr 7671  0_Rc0r 7672  ℝcr 8185  1c1 8187   + caddc 8189   <_ℝ cltrr 8190   · cmul 8191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-2o 6692  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7678  df-pli 7679  df-mi 7680  df-lti 7681  df-plpq 7718  df-mpq 7719  df-enq 7721  df-nqqs 7722  df-plqqs 7723  df-mqqs 7724  df-1nqqs 7725  df-rq 7726  df-ltnqqs 7727  df-enq0 7798  df-nq0 7799  df-0nq0 7800  df-plq0 7801  df-mq0 7802  df-inp 7840  df-i1p 7841  df-iplp 7842  df-enr 8100  df-nr 8101  df-plr 8102  df-0r 8105  df-1r 8106  df-c 8192  df-1 8194  df-r 8196  df-add 8197

This theorem is used by:  axcaucvglemcau  8272  axcaucvglemres  8273