Theorem nghmcn 25044

Description: A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)

Hypotheses

Ref	Expression
nghmcn.j	⊢ 𝐽 = (TopOpen‘𝑆)
nghmcn.k	⊢ 𝐾 = (TopOpen‘𝑇)

Assertion

Ref	Expression
nghmcn	⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem nghmcn

Dummy variables 𝑠 𝑟 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nghmghm 25033	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		eqid 2761	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2761	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 19414	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	1, 4	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		simprr 785	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
7		eqid 2761	. . . . . . . . 9 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
8	7	nghmcl 25026	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
9		nghmrcl1 25031	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
10		nghmrcl2 25032	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
11	7	nmoge0 25020	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
12	9, 10, 1, 11	syl3anc 1398	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
13	8, 12	ge0p1rpd 13175	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
14	13	adantr 486	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
15	6, 14	rpdivcld 13162	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ+)
16		ngpms 24899	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
17	9, 16	syl 18	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ MetSp)
18	17	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ MetSp)
19		simplrl 789	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simpr 490	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21		eqid 2761	. . . . . . . . . . 11 ⊢ (dist‘𝑆) = (dist‘𝑆)
22	2, 21	mscl 24760	. . . . . . . . . 10 ⊢ ((𝑆 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
23	18, 19, 20, 22	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
24	6	adantr 486	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ+)
25	24	rpred 13145	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ)
26	13	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
27	23, 25, 26	ltmuldiv2d 13193	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟 ↔ (𝑥(dist‘𝑆)𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1))))
28		ngpms 24899	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
29	10, 28	syl 18	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ MetSp)
30	29	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ MetSp)
31	5	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
32	31, 19	ffvelcdmd 7077	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑥) ∈ (Base‘𝑇))
33	31, 20	ffvelcdmd 7077	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑦) ∈ (Base‘𝑇))
34		eqid 2761	. . . . . . . . . . . 12 ⊢ (dist‘𝑇) = (dist‘𝑇)
35	3, 34	mscl 24760	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ MetSp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
36	30, 32, 33, 35	syl3anc 1398	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
37	8	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
38	37, 23	remulcld 11320	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ)
39	26	rpred 13145	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ)
40	39, 23	remulcld 11320	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ)
41	7, 2, 21, 34	nmods 25043	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
42	41	3expa 1136	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
43	42	adantlrr 734	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
44		msxms 24753	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
45	18, 44	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ ∞MetSp)
46	2, 21	xmsge0 24762	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
47	45, 19, 20, 46	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
48	37	lep1d 12229	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ≤ (((𝑆 normOp 𝑇)‘𝐹) + 1))
49	37, 39, 23, 47, 48	lemul1ad 12237	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)))
50	36, 38, 40, 43, 49	letrd 11448	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)))
51		lelttr 11381	. . . . . . . . . 10 ⊢ ((((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) < 𝑟))
52	36, 40, 25, 51	syl3anc 1398	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) < 𝑟))
53	50, 52	mpand 708	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟 → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) < 𝑟))
54	27, 53	sylbird 263	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑥(dist‘𝑆)𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) < 𝑟))
55	19, 20	ovresd 7579	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) = (𝑥(dist‘𝑆)𝑦))
56	55	breq1d 5113	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ↔ (𝑥(dist‘𝑆)𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1))))
57	32, 33	ovresd 7579	. . . . . . . 8 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) = ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)))
58	57	breq1d 5113	. . . . . . 7 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟 ↔ ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) < 𝑟))
59	54, 56, 58	3imtr4d 297	. . . . . 6 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))
60	59	ralrimiva 3155	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))
61		breq2 5107	. . . . . 6 ⊢ (𝑠 = (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 ↔ (𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1))))
62	61	rspceaimv 3583	. . . . 5 ⊢ (((𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ+ ∧ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟)) → ∃𝑠 ∈ ℝ+ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))
63	15, 60, 62	syl2anc 596	. . . 4 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → ∃𝑠 ∈ ℝ+ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))
64	63	ralrimivva 3206	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))
65		eqid 2761	. . . . . 6 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
66	2, 65	xmsxmet 24755	. . . . 5 ⊢ (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
67	17, 44, 66	3syl 19	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
68		msxms 24753	. . . . 5 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
69		eqid 2761	. . . . . 6 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
70	3, 69	xmsxmet 24755	. . . . 5 ⊢ (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
71	29, 68, 70	3syl 19	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
72		eqid 2761	. . . . 5 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
73		eqid 2761	. . . . 5 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
74	72, 73	metcn 24842	. . . 4 ⊢ ((((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)) ∧ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇))) → (𝐹 ∈ ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))) ↔ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))))
75	67, 71, 74	syl2anc 596	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝐹 ∈ ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))) ↔ (𝐹:(Base‘𝑆)⟶(Base‘𝑇) ∧ ∀𝑥 ∈ (Base‘𝑆)∀𝑟 ∈ ℝ+ ∃𝑠 ∈ ℝ+ ∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟))))
76	5, 64, 75	mpbir2and 726	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))))
77		nghmcn.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑆)
78	77, 2, 65	mstopn 24751	. . . 4 ⊢ (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
79	17, 78	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
80		nghmcn.k	. . . . 5 ⊢ 𝐾 = (TopOpen‘𝑇)
81	80, 3, 69	mstopn 24751	. . . 4 ⊢ (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
82	29, 81	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
83	79, 82	oveq12d 7430	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝐽 Cn 𝐾) = ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))))
84	76, 83	eleqtrrd 2864	1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   class class class wbr 5103   × cxp 5649   ↾ cres 5653  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  ℝcr 11180  0cc0 11181  1c1 11182   + caddc 11184   · cmul 11186   < clt 11324   ≤ cle 11325   / cdiv 11954  ℝ⁺crp 13101  Basecbs 17367  distcds 17417  TopOpenctopn 17572   GrpHom cghm 19407  ∞Metcxmet 21643  MetOpencmopn 21648   Cn ccn 23522  ∞MetSpcxms 24616  MetSpcms 24617  NrmGrpcngp 24876   normOp cnmo 25004   NGHom cnghm 25005

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-cnex 11237  ax-resscn 11238  ax-1cn 11239  ax-icn 11240  ax-addcl 11241  ax-addrcl 11242  ax-mulcl 11243  ax-mulrcl 11244  ax-mulcom 11245  ax-addass 11246  ax-mulass 11247  ax-distr 11248  ax-i2m1 11249  ax-1ne0 11250  ax-1rid 11251  ax-rnegex 11252  ax-rrecex 11253  ax-cnre 11254  ax-pre-lttri 11255  ax-pre-lttrn 11256  ax-pre-ltadd 11257  ax-pre-mulgt0 11258  ax-pre-sup 11259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-sup 9418  df-inf 9419  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11524  df-neg 11525  df-div 11955  df-nn 12317  df-2 12386  df-n0 12588  df-z 12675  df-uz 12947  df-q 13057  df-rp 13102  df-xneg 13222  df-xadd 13223  df-xmul 13224  df-ico 13463  df-0g 17592  df-topgen 17594  df-mgm 18796  df-sgrp 18888  df-mnd 18904  df-grp 19127  df-minusg 19128  df-sbg 19129  df-ghm 19408  df-psmet 21650  df-xmet 21651  df-met 21652  df-bl 21653  df-mopn 21654  df-top 23192  df-topon 23209  df-topsp 23231  df-bases 23244  df-cn 23525  df-cnp 23526  df-xms 24619  df-ms 24620  df-nm 24881  df-ngp 24882  df-nmo 25007  df-nghm 25008

This theorem is used by:  nmhmcn  25421