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Theorem metcnp 24457

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 24456	. 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 6654	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
5	4	ad2antlr 727	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → Fun 𝐹)
6		simpll1 1213	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐶 ∈ (∞Met‘𝑋))
7		simpll3 1215	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
8		rpxr 12900	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
9	8	ad2antll 729	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ^*)
10		blssm 24334	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
11	6, 7, 9, 10	syl3anc 1373	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
12		fdm 6660	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
13	12	ad2antlr 727	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → dom 𝐹 = 𝑋)
14	11, 13	sseqtrrd 3972	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ dom 𝐹)
15		funimass4 6886	. . . . . . . 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 584	. . . . . . 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 24304	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑤 ∈ (𝑃(ball‘𝐶)𝑧) ↔ (𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧)))
18	6, 7, 9, 17	syl3anc 1373	. . . . . . . . . 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 1193	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
22	21	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
23		simplrl 776	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ⁺)
24	23	rpxrd 12935	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ^*)
25		simpllr 775	. . . . . . . . . . . . . 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	ffvelcdmd 7018	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
28		simplr 768	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐹:𝑋⟶𝑌)
29	28	ffvelcdmda 7017	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) ∈ 𝑌)
30		elbl2 24306	. . . . . . . . . . . . 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 838	. . . . . . . . . . . 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 803	. . . . . . . . . 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	bitrid 283	. . . . . . . . 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 279	. . . . . . . 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 3151	. . . . . . 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 279	. . . . . 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 3154	. . . 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 3153	. . 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 579	. 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 279	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ⊆ wss 3902 class class class wbr 5091 dom cdm 5616 “ cima 5619 Fun wfun 6475 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℝ^*cxr 11145 < clt 11146 ℝ⁺crp 12890 ∞Metcxmet 21277 ballcbl 21279 MetOpencmopn 21282 CnP ccnp 23141

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-sup 9326 df-inf 9327 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-n0 12382 df-z 12469 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-topgen 17347 df-psmet 21284 df-xmet 21285 df-bl 21287 df-mopn 21288 df-top 22810 df-topon 22827 df-bases 22862 df-cnp 23144

This theorem is referenced by: metcnp2 24458 metcn 24459 metcnpi 24460 txmetcnp 24463 abelth 26379 qqhcn 34002

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