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Theorem metcnpi3 24055

Description: Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 24054 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)

**Hypotheses**
Ref	Expression
metcn.2	⊢ 𝐽 = (MetOpen‘𝐶)
metcn.4	⊢ 𝐾 = (MetOpen‘𝐷)

**Assertion**
Ref	Expression
metcnpi3	⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦 𝑥,𝑋,𝑦 𝑥,𝑌,𝑦 𝑥,𝐴,𝑦 𝑥,𝐶,𝑦 𝑥,𝐷,𝑦 𝑥,𝑃,𝑦

Proof of Theorem metcnpi3 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		metcn.2	. . 3 ⊢ 𝐽 = (MetOpen‘𝐶)
2		metcn.4	. . 3 ⊢ 𝐾 = (MetOpen‘𝐷)
3	1, 2	metcnpi2 24054	. 2 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) → ∃𝑧 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))
4		rphalfcl 13001	. . . 4 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) ∈ ℝ⁺)
5	4	ad2antrl 727	. . 3 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))) → (𝑧 / 2) ∈ ℝ⁺)
6		simplll 774	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐶 ∈ (∞Met‘𝑋))
7		simprr 772	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝑦 ∈ 𝑋)
8		simplrl 776	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃))
9		eqid 2733	. . . . . . . . . . . 12 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	cnprcl 22749	. . . . . . . . . . 11 ⊢ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) → 𝑃 ∈ ∪ 𝐽)
11	8, 10	syl 17	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝑃 ∈ ∪ 𝐽)
12	1	mopnuni 23947	. . . . . . . . . . 11 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
13	6, 12	syl 17	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝑋 = ∪ 𝐽)
14	11, 13	eleqtrrd 2837	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝑃 ∈ 𝑋)
15		xmetcl 23837	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑦𝐶𝑃) ∈ ℝ^*)
16	6, 7, 14, 15	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝑦𝐶𝑃) ∈ ℝ^*)
17	4	ad2antrl 727	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝑧 / 2) ∈ ℝ⁺)
18	17	rpxrd 13017	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝑧 / 2) ∈ ℝ^*)
19		rpxr 12983	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
20	19	ad2antrl 727	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝑧 ∈ ℝ^*)
21		rphalflt 13003	. . . . . . . . 9 ⊢ (𝑧 ∈ ℝ⁺ → (𝑧 / 2) < 𝑧)
22	21	ad2antrl 727	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝑧 / 2) < 𝑧)
23		xrlelttr 13135	. . . . . . . . . 10 ⊢ (((𝑦𝐶𝑃) ∈ ℝ^* ∧ (𝑧 / 2) ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → (((𝑦𝐶𝑃) ≤ (𝑧 / 2) ∧ (𝑧 / 2) < 𝑧) → (𝑦𝐶𝑃) < 𝑧))
24	23	expcomd 418	. . . . . . . . 9 ⊢ (((𝑦𝐶𝑃) ∈ ℝ^* ∧ (𝑧 / 2) ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) → ((𝑧 / 2) < 𝑧 → ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → (𝑦𝐶𝑃) < 𝑧)))
25	24	imp 408	. . . . . . . 8 ⊢ ((((𝑦𝐶𝑃) ∈ ℝ^* ∧ (𝑧 / 2) ∈ ℝ^* ∧ 𝑧 ∈ ℝ^*) ∧ (𝑧 / 2) < 𝑧) → ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → (𝑦𝐶𝑃) < 𝑧))
26	16, 18, 20, 22, 25	syl31anc 1374	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → (𝑦𝐶𝑃) < 𝑧))
27		simpllr 775	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐷 ∈ (∞Met‘𝑌))
28	1	mopntopon 23945	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
29	6, 28	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐽 ∈ (TopOn‘𝑋))
30	2	mopntopon 23945	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐾 ∈ (TopOn‘𝑌))
31	27, 30	syl 17	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐾 ∈ (TopOn‘𝑌))
32		cnpf2 22754	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃)) → 𝐹:𝑋⟶𝑌)
33	29, 31, 8, 32	syl3anc 1372	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐹:𝑋⟶𝑌)
34	33, 7	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑦) ∈ 𝑌)
35	33, 14	ffvelcdmd 7088	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (𝐹‘𝑃) ∈ 𝑌)
36		xmetcl 23837	. . . . . . . . 9 ⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑦) ∈ 𝑌 ∧ (𝐹‘𝑃) ∈ 𝑌) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ∈ ℝ^*)
37	27, 34, 35, 36	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ∈ ℝ^*)
38		simplrr 777	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ ℝ⁺)
39	38	rpxrd 13017	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → 𝐴 ∈ ℝ^*)
40		xrltle 13128	. . . . . . . 8 ⊢ ((((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → (((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
41	37, 39, 40	syl2anc 585	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
42	26, 41	imim12d 81	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ 𝑦 ∈ 𝑋)) → (((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴) → ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴)))
43	42	anassrs 469	. . . . 5 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ 𝑧 ∈ ℝ⁺) ∧ 𝑦 ∈ 𝑋) → (((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴) → ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴)))
44	43	ralimdva 3168	. . . 4 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ 𝑧 ∈ ℝ⁺) → (∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴) → ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴)))
45	44	impr 456	. . 3 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))) → ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
46		breq2 5153	. . . 4 ⊢ (𝑥 = (𝑧 / 2) → ((𝑦𝐶𝑃) ≤ 𝑥 ↔ (𝑦𝐶𝑃) ≤ (𝑧 / 2)))
47	46	rspceaimv 3618	. . 3 ⊢ (((𝑧 / 2) ∈ ℝ⁺ ∧ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ (𝑧 / 2) → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴)) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
48	5, 45, 47	syl2anc 585	. 2 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) ∧ (𝑧 ∈ ℝ⁺ ∧ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) < 𝑧 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) < 𝐴))) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))
49	3, 48	rexlimddv 3162	1 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ⁺)) → ∃𝑥 ∈ ℝ⁺ ∀𝑦 ∈ 𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹‘𝑦)𝐷(𝐹‘𝑃)) ≤ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∪ cuni 4909 class class class wbr 5149 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 / cdiv 11871 2c2 12267 ℝ⁺crp 12974 ∞Metcxmet 20929 MetOpencmopn 20934 TopOnctopon 22412 CnP ccnp 22729

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-bl 20939 df-mopn 20940 df-top 22396 df-topon 22413 df-bases 22449 df-cnp 22732

This theorem is referenced by: blocnilem 30057

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