Theorem axcaucvglemval 8208

Description: Lemma for axcaucvg 8211. 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 3882	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝐽 → ⟨𝑗, 1o⟩ = ⟨𝐽, 1o⟩)
4	3	eceq1d 6802	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝐽 → [⟨𝑗, 1o⟩] ~Q = [⟨𝐽, 1o⟩] ~Q )
5	4	breq2d 4120	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → (𝑙 <Q [⟨𝑗, 1o⟩] ~Q ↔ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q ))
6	5	abbidv 2352	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → {𝑙 ∣ 𝑙 <Q [⟨𝑗, 1o⟩] ~Q } = {𝑙 ∣ 𝑙 <Q [⟨𝐽, 1o⟩] ~Q })
7	4	breq1d 4118	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝐽 → ([⟨𝑗, 1o⟩] ~Q <Q 𝑢 ↔ [⟨𝐽, 1o⟩] ~Q <Q 𝑢))
8	7	abbidv 2352	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → {𝑢 ∣ [⟨𝑗, 1o⟩] ~Q <Q 𝑢} = {𝑢 ∣ [⟨𝐽, 1o⟩] ~Q <Q 𝑢})
9	6, 8	opeq12d 3890	. . . . . . . . . . . 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 6064	. . . . . . . . . . 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 3888	. . . . . . . . . 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 6802	. . . . . . . . 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 3888	. . . . . . . 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 5673	. . . . . . 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 2241	. . . . . 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 6004	. . . . 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 8206	. . . 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 5757	. . 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 2238	. 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 2309	. . 3 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → (𝐺‘𝐽) ∈ R)
25	20	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐽 ∈ N) → 𝐹:𝑁⟶ℝ)
26		pitonn 8159	. . . . . . . 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 2326	. . . . . . 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 5812	. . . . 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 8140	. . . . 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 2234	. . . . 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 2730	. . . 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 3882	. . . . 5 ⊢ (𝑧 = (𝐺‘𝐽) → ⟨𝑧, 0R⟩ = ⟨(𝐺‘𝐽), 0R⟩)
36	35	eqeq2d 2244	. . . 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 6026	. . 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 411	. 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⟩)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃!wreu 2522  ⟨cop 3691  ∩ cint 3948   class class class wbr 4108   ↦ cmpt 4170  ⟶wf 5347  ‘cfv 5351  ℩crio 6001  (class class class)co 6049  1_oc1o 6639  [cec 6764  Ncnpi 7583   ~_Q ceq 7590   <_Q cltq 7596  1_Pc1p 7603   +_P cpp 7604   ~_R cer 7607  Rcnr 7608  0_Rc0r 7609  ℝcr 8122  1c1 8124   + caddc 8126   <_ℝ cltrr 8127   · cmul 8128

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-eprel 4409  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-1o 6646  df-2o 6647  df-oadd 6650  df-omul 6651  df-er 6766  df-ec 6768  df-qs 6772  df-ni 7615  df-pli 7616  df-mi 7617  df-lti 7618  df-plpq 7655  df-mpq 7656  df-enq 7658  df-nqqs 7659  df-plqqs 7660  df-mqqs 7661  df-1nqqs 7662  df-rq 7663  df-ltnqqs 7664  df-enq0 7735  df-nq0 7736  df-0nq0 7737  df-plq0 7738  df-mq0 7739  df-inp 7777  df-i1p 7778  df-iplp 7779  df-enr 8037  df-nr 8038  df-plr 8039  df-0r 8042  df-1r 8043  df-c 8129  df-1 8131  df-r 8133  df-add 8134

This theorem is referenced by:  axcaucvglemcau  8209  axcaucvglemres  8210