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Theorem nghmcn 24253

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 24242	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2		eqid 2732	. . . . 5 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqid 2732	. . . . 5 ⊢ (Base‘𝑇) = (Base‘𝑇)
4	2, 3	ghmf 19090	. . . 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 771	. . . . . 6 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) → 𝑟 ∈ ℝ⁺)
7		eqid 2732	. . . . . . . . 9 ⊢ (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
8	7	nghmcl 24235	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
9		nghmrcl1 24240	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
10		nghmrcl2 24241	. . . . . . . . 9 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
11	7	nmoge0 24229	. . . . . . . . 9 ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
12	9, 10, 1, 11	syl3anc 1371	. . . . . . . 8 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
13	8, 12	ge0p1rpd 13042	. . . . . . 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 13029	. . . . 5 ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) → (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ⁺)
16		ngpms 24100	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
17	9, 16	syl 17	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ MetSp)
18	17	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ MetSp)
19		simplrl 775	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20		simpr 485	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21		eqid 2732	. . . . . . . . . . 11 ⊢ (dist‘𝑆) = (dist‘𝑆)
22	2, 21	mscl 23958	. . . . . . . . . 10 ⊢ ((𝑆 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
23	18, 19, 20, 22	syl3anc 1371	. . . . . . . . 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 13012	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ)
26	13	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ⁺)
27	23, 25, 26	ltmuldiv2d 13060	. . . . . . . 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 24100	. . . . . . . . . . . . 13 ⊢ (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
29	10, 28	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ MetSp)
30	29	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ MetSp)
31	5	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
32	31, 19	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑥) ∈ (Base‘𝑇))
33	31, 20	ffvelcdmd 7084	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹‘𝑦) ∈ (Base‘𝑇))
34		eqid 2732	. . . . . . . . . . . 12 ⊢ (dist‘𝑇) = (dist‘𝑇)
35	3, 34	mscl 23958	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ MetSp ∧ (𝐹‘𝑥) ∈ (Base‘𝑇) ∧ (𝐹‘𝑦) ∈ (Base‘𝑇)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
36	30, 32, 33, 35	syl3anc 1371	. . . . . . . . . 10 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹‘𝑥)(dist‘𝑇)(𝐹‘𝑦)) ∈ ℝ)
37	8	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
38	37, 23	remulcld 11240	. . . . . . . . . 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 13012	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ)
40	39, 23	remulcld 11240	. . . . . . . . . 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 24252	. . . . . . . . . . . 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 1118	. . . . . . . . . . 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 719	. . . . . . . . . 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 23951	. . . . . . . . . . . . 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 23960	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
47	45, 19, 20, 46	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ⁺)) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
48	37	lep1d 12141	. . . . . . . . . . 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 12149	. . . . . . . . . 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 11367	. . . . . . . . 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 11300	. . . . . . . . . 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 1371	. . . . . . . . 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 693	. . . . . . . 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 259	. . . . . . 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 7570	. . . . . . . 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 5157	. . . . . . 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 7570	. . . . . . . 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 5157	. . . . . . 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 293	. . . . . 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 3146	. . . . 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 5151	. . . . . 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 3616	. . . . 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 584	. . . 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 3200	. . 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 2732	. . . . . 6 ⊢ ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
66	2, 65	xmsxmet 23953	. . . . 5 ⊢ (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
67	17, 44, 66	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
68		msxms 23951	. . . . 5 ⊢ (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
69		eqid 2732	. . . . . 6 ⊢ ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
70	3, 69	xmsxmet 23953	. . . . 5 ⊢ (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
71	29, 68, 70	3syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
72		eqid 2732	. . . . 5 ⊢ (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
73		eqid 2732	. . . . 5 ⊢ (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
74	72, 73	metcn 24043	. . . 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 584	. . 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 711	. 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 23949	. . . 4 ⊢ (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
79	17, 78	syl 17	. . 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 23949	. . . 4 ⊢ (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
82	29, 81	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
83	79, 82	oveq12d 7423	. 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 2836	1 ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 class class class wbr 5147 × cxp 5673 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 / cdiv 11867 ℝ⁺crp 12970 Basecbs 17140 distcds 17202 TopOpenctopn 17363 GrpHom cghm 19083 ∞Metcxmet 20921 MetOpencmopn 20926 Cn ccn 22719 ∞MetSpcxms 23814 MetSpcms 23815 NrmGrpcngp 24077 normOp cnmo 24213 NGHom cnghm 24214

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ico 13326 df-0g 17383 df-topgen 17385 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-ghm 19084 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cn 22722 df-cnp 22723 df-xms 23817 df-ms 23818 df-nm 24082 df-ngp 24083 df-nmo 24216 df-nghm 24217

This theorem is referenced by: nmhmcn 24627

W3C validator