metcnp - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > metcnp		GIF version

Theorem metcnp 13592

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 13591	. 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 5360	. . . . . . . . 9 ⊢ (𝐹:𝑋⟶𝑌 → Fun 𝐹)
5	4	ad2antlr 489	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → Fun 𝐹)
6		simpll1 1036	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐶 ∈ (∞Met‘𝑋))
7		simpll3 1038	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑃 ∈ 𝑋)
8		rpxr 9632	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ℝ⁺ → 𝑧 ∈ ℝ^*)
9	8	ad2antll 491	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝑧 ∈ ℝ^*)
10		blssm 13501	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
11	6, 7, 9, 10	syl3anc 1238	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ 𝑋)
12		fdm 5363	. . . . . . . . . 10 ⊢ (𝐹:𝑋⟶𝑌 → dom 𝐹 = 𝑋)
13	12	ad2antlr 489	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → dom 𝐹 = 𝑋)
14	11, 13	sseqtrrd 3192	. . . . . . . 8 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → (𝑃(ball‘𝐶)𝑧) ⊆ dom 𝐹)
15		funimass4 5558	. . . . . . . 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 411	. . . . . . 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 13471	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑧 ∈ ℝ^*) → (𝑤 ∈ (𝑃(ball‘𝐶)𝑧) ↔ (𝑤 ∈ 𝑋 ∧ (𝑃𝐶𝑤) < 𝑧)))
18	6, 7, 9, 17	syl3anc 1238	. . . . . . . . . 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 231	. . . . . . . . 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 263	. . . . . . . . . 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 1001	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) → 𝐷 ∈ (∞Met‘𝑌))
22	21	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
23		simplrl 535	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ⁺)
24	23	rpxrd 9668	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑦 ∈ ℝ^*)
25		simpllr 534	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝐹:𝑋⟶𝑌)
26	7	adantr 276	. . . . . . . . . . . . . 14 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → 𝑃 ∈ 𝑋)
27	25, 26	ffvelcdmd 5644	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
28		simplr 528	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) → 𝐹:𝑋⟶𝑌)
29	28	ffvelcdmda 5643	. . . . . . . . . . . . 13 ⊢ (((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃 ∈ 𝑋) ∧ 𝐹:𝑋⟶𝑌) ∧ (𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ ℝ⁺)) ∧ 𝑤 ∈ 𝑋) → (𝐹‘𝑤) ∈ 𝑌)
30		elbl2 13473	. . . . . . . . . . . . 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 1239	. . . . . . . . . . . 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 230	. . . . . . . . . . 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 443	. . . . . . . . . 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 192	. . . . . . . . 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 188	. . . . . . . 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 2477	. . . . . . 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 188	. . . . . 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 400	. . . . 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 2472	. . . 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 2471	. . 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 452	. 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 188	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 set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2146 ∀wral 2453 ∃wrex 2454 ⊆ wss 3127 class class class wbr 3998 dom cdm 4620 “ cima 4623 Fun wfun 5202 ⟶wf 5204 ‘cfv 5208 (class class class)co 5865 ℝ^*cxr 7965 < clt 7966 ℝ⁺crp 9624 ∞Metcxmet 13060 ballcbl 13062 MetOpencmopn 13065 CnP ccnp 13266

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906

This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-map 6640 df-sup 6973 df-inf 6974 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-xneg 9743 df-xadd 9744 df-seqfrec 10416 df-exp 10490 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-topgen 12640 df-psmet 13067 df-xmet 13068 df-bl 13070 df-mopn 13071 df-top 13076 df-topon 13089 df-bases 13121 df-cnp 13269

This theorem is referenced by: metcnp2 13593 metcn 13594 metcnpi 13595 txmetcnp 13598 metcnpd 13600

W3C validator