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Theorem nghmcn 24766
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.
StepHypRef Expression
1 nghmghm 24755 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 eqid 2737 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2737 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 19238 . . . 4 (𝐹 ∈ (𝑆 GrpHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
51, 4syl 17 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
6 simprr 773 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
7 eqid 2737 . . . . . . . . 9 (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
87nghmcl 24748 . . . . . . . 8 (𝐹 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
9 nghmrcl1 24753 . . . . . . . . 9 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
10 nghmrcl2 24754 . . . . . . . . 9 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
117nmoge0 24742 . . . . . . . . 9 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
129, 10, 1, 11syl3anc 1373 . . . . . . . 8 (𝐹 ∈ (𝑆 NGHom 𝑇) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
138, 12ge0p1rpd 13107 . . . . . . 7 (𝐹 ∈ (𝑆 NGHom 𝑇) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
1413adantr 480 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
156, 14rpdivcld 13094 . . . . 5 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ+)
16 ngpms 24613 . . . . . . . . . . . 12 (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
179, 16syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ MetSp)
1817ad2antrr 726 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ MetSp)
19 simplrl 777 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑥 ∈ (Base‘𝑆))
20 simpr 484 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑦 ∈ (Base‘𝑆))
21 eqid 2737 . . . . . . . . . . 11 (dist‘𝑆) = (dist‘𝑆)
222, 21mscl 24471 . . . . . . . . . 10 ((𝑆 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
2318, 19, 20, 22syl3anc 1373 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
246adantr 480 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ+)
2524rpred 13077 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ)
2613ad2antrr 726 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
2723, 25, 26ltmuldiv2d 13125 . . . . . . . 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 24613 . . . . . . . . . . . . 13 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
2910, 28syl 17 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ MetSp)
3029ad2antrr 726 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ MetSp)
315ad2antrr 726 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3231, 19ffvelcdmd 7105 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹𝑥) ∈ (Base‘𝑇))
3331, 20ffvelcdmd 7105 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹𝑦) ∈ (Base‘𝑇))
34 eqid 2737 . . . . . . . . . . . 12 (dist‘𝑇) = (dist‘𝑇)
353, 34mscl 24471 . . . . . . . . . . 11 ((𝑇 ∈ MetSp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ∈ ℝ)
3630, 32, 33, 35syl3anc 1373 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ∈ ℝ)
378ad2antrr 726 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
3837, 23remulcld 11291 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ)
3926rpred 13077 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ)
4039, 23remulcld 11291 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ)
417, 2, 21, 34nmods 24765 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
42413expa 1119 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
4342adantlrr 721 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
44 msxms 24464 . . . . . . . . . . . . 13 (𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp)
4518, 44syl 17 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑆 ∈ ∞MetSp)
462, 21xmsge0 24473 . . . . . . . . . . . 12 ((𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
4745, 19, 20, 46syl3anc 1373 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
4837lep1d 12199 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ≤ (((𝑆 normOp 𝑇)‘𝐹) + 1))
4937, 39, 23, 47, 48lemul1ad 12207 . . . . . . . . . 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)))
5036, 38, 40, 43, 49letrd 11418 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)))
51 lelttr 11351 . . . . . . . . . 10 ((((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ∈ ℝ ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ ∧ 𝑟 ∈ ℝ) → ((((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) < 𝑟))
5236, 40, 25, 51syl3anc 1373 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) ∧ ((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) < 𝑟))
5350, 52mpand 695 . . . . . . . 8 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((((𝑆 normOp 𝑇)‘𝐹) + 1) · (𝑥(dist‘𝑆)𝑦)) < 𝑟 → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) < 𝑟))
5427, 53sylbird 260 . . . . . . 7 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑥(dist‘𝑆)𝑦) < (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) < 𝑟))
5519, 20ovresd 7600 . . . . . . . 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))
5655breq1d 5153 . . . . . . 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))))
5732, 33ovresd 7600 . . . . . . . 8 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) = ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)))
5857breq1d 5153 . . . . . . 7 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟 ↔ ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) < 𝑟))
5954, 56, 583imtr4d 294 . . . . . 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)) → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
6059ralrimiva 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)) → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
61 breq2 5147 . . . . . 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))))
6261rspceaimv 3628 . . . . 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)) → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟)) → ∃𝑠 ∈ ℝ+𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
6315, 60, 62syl2anc 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 → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
6463ralrimivva 3202 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → ∀𝑥 ∈ (Base‘𝑆)∀𝑟 ∈ ℝ+𝑠 ∈ ℝ+𝑦 ∈ (Base‘𝑆)((𝑥((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))𝑦) < 𝑠 → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
65 eqid 2737 . . . . . 6 ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
662, 65xmsxmet 24466 . . . . 5 (𝑆 ∈ ∞MetSp → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
6717, 44, 663syl 18 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) ∈ (∞Met‘(Base‘𝑆)))
68 msxms 24464 . . . . 5 (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
69 eqid 2737 . . . . . 6 ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
703, 69xmsxmet 24466 . . . . 5 (𝑇 ∈ ∞MetSp → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
7129, 68, 703syl 18 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) ∈ (∞Met‘(Base‘𝑇)))
72 eqid 2737 . . . . 5 (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
73 eqid 2737 . . . . 5 (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
7472, 73metcn 24556 . . . 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 → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))))
7567, 71, 74syl2anc 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 → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))))
765, 64, 75mpbir2and 713 . 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)
7877, 2, 65mstopn 24462 . . . 4 (𝑆 ∈ MetSp → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
7917, 78syl 17 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐽 = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))))
80 nghmcn.k . . . . 5 𝐾 = (TopOpen‘𝑇)
8180, 3, 69mstopn 24462 . . . 4 (𝑇 ∈ MetSp → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
8229, 81syl 17 . . 3 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐾 = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))))
8379, 82oveq12d 7449 . 2 (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝐽 Cn 𝐾) = ((MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) Cn (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))))
8476, 83eleqtrrd 2844 1 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070   class class class wbr 5143   × cxp 5683  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296   / cdiv 11920  +crp 13034  Basecbs 17247  distcds 17306  TopOpenctopn 17466   GrpHom cghm 19230  ∞Metcxmet 21349  MetOpencmopn 21354   Cn ccn 23232  ∞MetSpcxms 24327  MetSpcms 24328  NrmGrpcngp 24590   normOp cnmo 24726   NGHom cnghm 24727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-sup 9482  df-inf 9483  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ico 13393  df-0g 17486  df-topgen 17488  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-ghm 19231  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cn 23235  df-cnp 23236  df-xms 24330  df-ms 24331  df-nm 24595  df-ngp 24596  df-nmo 24729  df-nghm 24730
This theorem is referenced by:  nmhmcn  25153
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