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Theorem metcnp 23603

Description: Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

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

**Assertion**
Ref	Expression
metcnp	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))

Distinct variable groups: 𝑦,𝑤,𝑧,𝐹 𝑤,𝐽,𝑦,𝑧 𝑤,𝐾,𝑦,𝑧 𝑤,𝑋,𝑦,𝑧 𝑤,𝑌,𝑦,𝑧 𝑤,𝐶,𝑦,𝑧 𝑤,𝐷,𝑦,𝑧 𝑤,𝑃,𝑦,𝑧

**Proof of Theorem metcnp**
Step	Hyp	Ref	Expression
1		metcn.2	. . 3 ⊢ 𝐽 = (MetOpen‘𝐶)
2		metcn.4	. . 3 ⊢ 𝐾 = (MetOpen‘𝐷)
3	1, 2	metcnp3 23602	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
4		ffun 6587	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
5	4	ad2antlr 723	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → Fun 𝐹)
6		simpll1 1210	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐶 ∈ (∞Met‘𝑋))
7		simpll3 1212	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
8		rpxr 12668	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
9	8	ad2antll 725	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ^*)
10		blssm 23479	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
11	6, 7, 9, 10	syl3anc 1369	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
12		fdm 6593	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
13	12	ad2antlr 723	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → dom 𝐹 = 𝑋)
14	11, 13	sseqtrrd 3958	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ dom 𝐹)
15		funimass4 6816	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (𝑃(ball‘𝐶)𝑧) ⊆ dom 𝐹) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑤 ∈ (𝑃(ball‘𝐶)𝑧)(𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
16	5, 14, 15	syl2anc 583	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑤 ∈ (𝑃(ball‘𝐶)𝑧)(𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)))
17		elbl 23449	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑤 ∈ (𝑃(ball‘𝐶)𝑧) ↔ (𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧)))
18	6, 7, 9, 17	syl3anc 1369	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑤 ∈ (𝑃(ball‘𝐶)𝑧) ↔ (𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧)))
19	18	imbi1d 341	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ((𝑤 ∈ (𝑃(ball‘𝐶)𝑧) → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ ((𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧) → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
20		impexp 450	. . . . . . . . . 10 ⊢ (((𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧) → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ (𝑤 ∈ 𝑋 → ((𝑃𝐶𝑤) < 𝑧 → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))))
21		simpl2 1190	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
22	21	ad2antrr 722	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
23		simplrl 773	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ⁺)
24	23	rpxrd 12702	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ^*)
25		simpllr 772	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
26	7	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑃 ∈ 𝑋)
27	25, 26	ffvelrnd 6944	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
28		simplr 765	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐹:𝑋⟶𝑌)
29	28	ffvelrnda 6943	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) ∈ 𝑌)
30		elbl2 23451	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑌) ∧ 𝑦 ∈ ℝ^*) ∧ ((𝐹‘𝑃) ∈ 𝑌 ∧ (𝐹‘𝑤) ∈ 𝑌)) → ((𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))
31	22, 24, 27, 29, 30	syl22anc 835	. . . . . . . . . . . 12 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → ((𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))
32	31	imbi2d 340	. . . . . . . . . . 11 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (((𝑃𝐶𝑤) < 𝑧 → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
33	32	pm5.74da 800	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ((𝑤 ∈ 𝑋 → ((𝑃𝐶𝑤) < 𝑧 → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦))) ↔ (𝑤 ∈ 𝑋 → ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
34	20, 33	syl5bb 282	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (((𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧) → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ (𝑤 ∈ 𝑋 → ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
35	19, 34	bitrd 278	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ((𝑤 ∈ (𝑃(ball‘𝐶)𝑧) → (𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ (𝑤 ∈ 𝑋 → ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
36	35	ralbidv2 3118	. . . . . . 7 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (∀𝑤 ∈ (𝑃(ball‘𝐶)𝑧)(𝐹‘𝑤) ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
37	16, 36	bitrd 278	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
38	37	anassrs 467	. . . . 5 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ ℝ⁺) → ((𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
39	38	rexbidva 3224	. . . 4 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
40	39	ralbidva 3119	. . 3 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦) ↔ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦)))
41	40	pm5.32da 578	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹‘𝑃)(ball‘𝐷)𝑦)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))
42	3, 41	bitrd 278	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑦 ∈ ℝ⁺ ∃𝑧 ∈ ℝ⁺ ∀𝑤 ∈ 𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹‘𝑃)𝐷(𝐹‘𝑤)) < 𝑦))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 ⊆ wss 3883 class class class wbr 5070 dom cdm 5580 “ cima 5583 Fun wfun 6412 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝ^*cxr 10939 < clt 10940 ℝ⁺crp 12659 ∞Metcxmet 20495 ballcbl 20497 MetOpencmopn 20500 CnP ccnp 22284

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-n0 12164 df-z 12250 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-bl 20505 df-mopn 20506 df-top 21951 df-topon 21968 df-bases 22004 df-cnp 22287

This theorem is referenced by: metcnp2 23604 metcn 23605 metcnpi 23606 txmetcnp 23609 abelth 25505 qqhcn 31841

W3C validator