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Theorem iscmet3lem1 25039

Description: Lemma for iscmet3 25041. (Contributed by Mario Carneiro, 15-Oct-2015.)

**Hypotheses**
Ref	Expression
iscmet3.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
iscmet3.2	⊢ 𝐽 = (MetOpen‘𝐷)
iscmet3.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
iscmet3.4	⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
iscmet3.6	⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
iscmet3.9	⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
iscmet3.10	⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))

**Assertion**
Ref	Expression
iscmet3lem1	⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))

Distinct variable groups: 𝑘,𝑛,𝑢,𝑣,𝐷 𝑘,𝐹,𝑛,𝑢,𝑣 𝑘,𝑋,𝑛 𝑘,𝐽,𝑛 𝑆,𝑘,𝑛,𝑢,𝑣 𝑘,𝑍,𝑛 𝑘,𝑀,𝑛 𝜑,𝑘,𝑛 Allowed substitution hints: 𝜑(𝑣,𝑢) 𝐽(𝑣,𝑢) 𝑀(𝑣,𝑢) 𝑋(𝑣,𝑢) 𝑍(𝑣,𝑢)

Proof of Theorem iscmet3lem1 Dummy variables 𝑗 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscmet3.3	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		iscmet3.1	. . . . . . 7 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	2	iscmet3lem3 25038	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑟 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((1 / 2)↑𝑘) < 𝑟)
4	1, 3	sylan 578	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((1 / 2)↑𝑘) < 𝑟)
5	2	r19.2uz 15302	. . . . 5 ⊢ (∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((1 / 2)↑𝑘) < 𝑟 → ∃𝑘 ∈ 𝑍 ((1 / 2)↑𝑘) < 𝑟)
6	4, 5	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑘 ∈ 𝑍 ((1 / 2)↑𝑘) < 𝑟)
7		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → (𝑆‘𝑛) = (𝑆‘𝑘))
8	7	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑘) ∈ (𝑆‘𝑛) ↔ (𝐹‘𝑘) ∈ (𝑆‘𝑘)))
9		iscmet3.10	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))
10	9	ad2antrr 722	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))
11		simpl 481	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘)) → 𝑘 ∈ 𝑍)
12	11	adantl 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑘 ∈ 𝑍)
13		rsp 3242	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛) → (𝑘 ∈ 𝑍 → ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛)))
14	10, 12, 13	sylc 65	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛))
15	12, 2	eleqtrdi 2841	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑘 ∈ (ℤ_≥‘𝑀))
16		eluzfz2 13513	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ (𝑀...𝑘))
17	15, 16	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑘 ∈ (𝑀...𝑘))
18	8, 14, 17	rspcdva 3612	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → (𝐹‘𝑘) ∈ (𝑆‘𝑘))
19	7	eleq2d 2817	. . . . . . . . . 10 ⊢ (𝑛 = 𝑘 → ((𝐹‘𝑗) ∈ (𝑆‘𝑛) ↔ (𝐹‘𝑗) ∈ (𝑆‘𝑘)))
20		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → (𝑀...𝑘) = (𝑀...𝑗))
21		fveq2 6890	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
22	21	eleq1d 2816	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ (𝑆‘𝑛) ↔ (𝐹‘𝑗) ∈ (𝑆‘𝑛)))
23	20, 22	raleqbidv 3340	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑗 → (∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛) ↔ ∀𝑛 ∈ (𝑀...𝑗)(𝐹‘𝑗) ∈ (𝑆‘𝑛)))
24	2	uztrn2 12845	. . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘)) → 𝑗 ∈ 𝑍)
25	24	adantl 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑗 ∈ 𝑍)
26	23, 10, 25	rspcdva 3612	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ∀𝑛 ∈ (𝑀...𝑗)(𝐹‘𝑗) ∈ (𝑆‘𝑛))
27		simprr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑗 ∈ (ℤ_≥‘𝑘))
28		elfzuzb 13499	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀...𝑗) ↔ (𝑘 ∈ (ℤ_≥‘𝑀) ∧ 𝑗 ∈ (ℤ_≥‘𝑘)))
29	15, 27, 28	sylanbrc 581	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑘 ∈ (𝑀...𝑗))
30	19, 26, 29	rspcdva 3612	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → (𝐹‘𝑗) ∈ (𝑆‘𝑘))
31		iscmet3.9	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
32	31	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
33		eluzelz 12836	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
34	33, 2	eleq2s 2849	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
35	34	ad2antrl 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑘 ∈ ℤ)
36		rsp 3242	. . . . . . . . . 10 ⊢ (∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘) → (𝑘 ∈ ℤ → ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)))
37	32, 35, 36	sylc 65	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))
38		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑢 = (𝐹‘𝑘) → (𝑢𝐷𝑣) = ((𝐹‘𝑘)𝐷𝑣))
39	38	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘𝑘) → ((𝑢𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ((𝐹‘𝑘)𝐷𝑣) < ((1 / 2)↑𝑘)))
40		oveq2 7419	. . . . . . . . . . 11 ⊢ (𝑣 = (𝐹‘𝑗) → ((𝐹‘𝑘)𝐷𝑣) = ((𝐹‘𝑘)𝐷(𝐹‘𝑗)))
41	40	breq1d 5157	. . . . . . . . . 10 ⊢ (𝑣 = (𝐹‘𝑗) → (((𝐹‘𝑘)𝐷𝑣) < ((1 / 2)↑𝑘) ↔ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < ((1 / 2)↑𝑘)))
42	39, 41	rspc2va 3622	. . . . . . . . 9 ⊢ ((((𝐹‘𝑘) ∈ (𝑆‘𝑘) ∧ (𝐹‘𝑗) ∈ (𝑆‘𝑘)) ∧ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < ((1 / 2)↑𝑘))
43	18, 30, 37, 42	syl21anc 834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < ((1 / 2)↑𝑘))
44		iscmet3.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋))
45	44	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝐷 ∈ (Met‘𝑋))
46		iscmet3.6	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶𝑋)
47	46	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → 𝐹:𝑍⟶𝑋)
48		ffvelcdm 7082	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶𝑋 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ 𝑋)
49	47, 11, 48	syl2an 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → (𝐹‘𝑘) ∈ 𝑋)
50		ffvelcdm 7082	. . . . . . . . . . 11 ⊢ ((𝐹:𝑍⟶𝑋 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ 𝑋)
51	47, 24, 50	syl2an 594	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → (𝐹‘𝑗) ∈ 𝑋)
52		metcl 24058	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (Met‘𝑋) ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ (𝐹‘𝑗) ∈ 𝑋) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ)
53	45, 49, 51, 52	syl3anc 1369	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ)
54		1rp 12982	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ⁺
55		rphalfcl 13005	. . . . . . . . . . . 12 ⊢ (1 ∈ ℝ⁺ → (1 / 2) ∈ ℝ⁺)
56	54, 55	ax-mp 5	. . . . . . . . . . 11 ⊢ (1 / 2) ∈ ℝ⁺
57		rpexpcl 14050	. . . . . . . . . . 11 ⊢ (((1 / 2) ∈ ℝ⁺ ∧ 𝑘 ∈ ℤ) → ((1 / 2)↑𝑘) ∈ ℝ⁺)
58	56, 35, 57	sylancr 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ((1 / 2)↑𝑘) ∈ ℝ⁺)
59	58	rpred 13020	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ((1 / 2)↑𝑘) ∈ ℝ)
60		rpre 12986	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ)
61	60	ad2antlr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → 𝑟 ∈ ℝ)
62		lttr 11294	. . . . . . . . 9 ⊢ ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) ∈ ℝ ∧ ((1 / 2)↑𝑘) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < ((1 / 2)↑𝑘) ∧ ((1 / 2)↑𝑘) < 𝑟) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
63	53, 59, 61, 62	syl3anc 1369	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → ((((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < ((1 / 2)↑𝑘) ∧ ((1 / 2)↑𝑘) < 𝑟) → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
64	43, 63	mpand 691	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ (𝑘 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑘))) → (((1 / 2)↑𝑘) < 𝑟 → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
65	64	anassrs 466	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑘)) → (((1 / 2)↑𝑘) < 𝑟 → ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
66	65	ralrimdva 3152	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ⁺) ∧ 𝑘 ∈ 𝑍) → (((1 / 2)↑𝑘) < 𝑟 → ∀𝑗 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
67	66	reximdva 3166	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → (∃𝑘 ∈ 𝑍 ((1 / 2)↑𝑘) < 𝑟 → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
68	6, 67	mpd 15	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ⁺) → ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟)
69	68	ralrimiva 3144	. 2 ⊢ (𝜑 → ∀𝑟 ∈ ℝ⁺ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟)
70		metxmet 24060	. . . 4 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
71	44, 70	syl 17	. . 3 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
72		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = (𝐹‘𝑗))
73		eqidd 2731	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = (𝐹‘𝑘))
74	2, 71, 1, 72, 73, 46	iscauf 25028	. 2 ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑟 ∈ ℝ⁺ ∃𝑘 ∈ 𝑍 ∀𝑗 ∈ (ℤ_≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑟))
75	69, 74	mpbird 256	1 ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 class class class wbr 5147 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 1c1 11113 < clt 11252 / cdiv 11875 2c2 12271 ℤcz 12562 ℤ_≥cuz 12826 ℝ⁺crp 12978 ...cfz 13488 ↑cexp 14031 ∞Metcxmet 21129 Metcmet 21130 MetOpencmopn 21134 Cauccau 25001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-inf 9440 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12979 df-xneg 13096 df-xadd 13097 df-fz 13489 df-fl 13761 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-psmet 21136 df-xmet 21137 df-met 21138 df-bl 21139 df-cau 25004

This theorem is referenced by: iscmet3 25041

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