Theorem nghmcn 22549

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 22538	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		eqid 2622	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2622	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 17664	. . . 4 ⊢ (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
5	1, 4	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6		simprr 796	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
7		eqid 2622	. . . . . . . . 9 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
8	7	nghmcl 22531	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
9		nghmrcl1 22536	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
10		nghmrcl2 22537	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
11	7	nmoge0 22525	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
12	9, 10, 1, 11	syl3anc 1326	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
13	8, 12	ge0p1rpd 11902	. . . . . . 7 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
14	13	adantr 481	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
15	6, 14	rpdivcld 11889	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ+)
16		ngpms 22404	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
17	9, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ MetSp)
18	17	ad2antrr 762	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ MetSp)
19		simplrl 800	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simpr 477	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21		eqid 2622	. . . . . . . . . . 11 ⊢ (dist‘𝑆) = (dist‘𝑆)
22	2, 21	mscl 22266	. . . . . . . . . 10 ⊢ ((𝑆 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
23	18, 19, 20, 22	syl3anc 1326	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
24	6	adantr 481	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ+)
25	24	rpred 11872	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ)
26	13	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
27	23, 25, 26	ltmuldiv2d 11920	. . . . . . . 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 22404	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
29	10, 28	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ MetSp)
30	29	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ MetSp)
31	5	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
32	31, 19	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑥) ∈ (Base‘𝑇))
33	31, 20	ffvelrnd 6360	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑦) ∈ (Base‘𝑇))
34		eqid 2622	. . . . . . . . . . . 12 ⊢ (dist‘𝑇) = (dist‘𝑇)
35	3, 34	mscl 22266	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ MetSp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
36	30, 32, 33, 35	syl3anc 1326	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
37	8	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
38	37, 23	remulcld 10070	. . . . . . . . . 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 11872	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ)
40	39, 23	remulcld 10070	. . . . . . . . . 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 22548	. . . . . . . . . . . 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 1265	. . . . . . . . . . 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 757	. . . . . . . . . 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 22259	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
45	18, 44	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ ∞MetSp)
46	2, 21	xmsge0 22268	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
47	45, 19, 20, 46	syl3anc 1326	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
48	37	lep1d 10955	. . . . . . . . . . 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 10963	. . . . . . . . . 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 10194	. . . . . . . . 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 10128	. . . . . . . . . 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 1326	. . . . . . . . 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 711	. . . . . . . 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 250	. . . . . . 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 6801	. . . . . . . 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 4663	. . . . . . 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 6801	. . . . . . . 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 4663	. . . . . . 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 283	. . . . . 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 2966	. . . . 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 4657	. . . . . . . 8 ⊢ (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	imbi1d 331	. . . . . . 7 ⊢ (𝑠 = (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → (((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 𝑟) ↔ ((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)))
63	62	ralbidv 2986	. . . . . 6 ⊢ (𝑠 = (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → (∀𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹‘𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹‘𝑦)) < 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)))
64	63	rspcev 3309	. . . . 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))
65	15, 60, 64	syl2anc 693	. . . 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))
66	65	ralrimivva 2971	. . 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))
67		eqid 2622	. . . . . 6 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
68	2, 67	xmsxmet 22261	. . . . 5 ⊢ (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
69	17, 44, 68	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
70		msxms 22259	. . . . 5 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
71		eqid 2622	. . . . . 6 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
72	3, 71	xmsxmet 22261	. . . . 5 ⊢ (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
73	29, 70, 72	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
74		eqid 2622	. . . . 5 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
75		eqid 2622	. . . . 5 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
76	74, 75	metcn 22348	. . . 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))))
77	69, 73, 76	syl2anc 693	. . 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))))
78	5, 66, 77	mpbir2and 957	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))))
79		nghmcn.j	. . . . 5 ⊢ 𝐽 = (TopOpen‘𝑆)
80	79, 2, 67	mstopn 22257	. . . 4 ⊢ (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
81	17, 80	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
82		nghmcn.k	. . . . 5 ⊢ 𝐾 = (TopOpen‘𝑇)
83	82, 3, 71	mstopn 22257	. . . 4 ⊢ (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
84	29, 83	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
85	81, 84	oveq12d 6668	. 2 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝐽 Cn 𝐾) = ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))))
86	78, 85	eleqtrrd 2704	1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   class class class wbr 4653   × cxp 5112   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   + caddc 9939   · cmul 9941   < clt 10074   ≤ cle 10075   / cdiv 10684  ℝ⁺crp 11832  Basecbs 15857  distcds 15950  TopOpenctopn 16082   GrpHom cghm 17657  ∞Metcxmt 19731  MetOpencmopn 19736   Cn ccn 21028  ∞MetSpcxme 22122  MetSpcmt 22123  NrmGrpcngp 22382   normOp cnmo 22509   NGHom cnghm 22510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-0g 16102  df-topgen 16104  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-xms 22125  df-ms 22126  df-nm 22387  df-ngp 22388  df-nmo 22512  df-nghm 22513

This theorem is referenced by:  nmhmcn  22920