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Theorem nghmcn 24109
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 24098 . . . 4 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
2 eqid 2736 . . . . 5 (Base‘𝑆) = (Base‘𝑆)
3 eqid 2736 . . . . 5 (Base‘𝑇) = (Base‘𝑇)
42, 3ghmf 19012 . . . 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 771 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ+)
7 eqid 2736 . . . . . . . . 9 (𝑆 normOp 𝑇) = (𝑆 normOp 𝑇)
87nghmcl 24091 . . . . . . . 8 (𝐹 ∈ (𝑆 NGHom 𝑇) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
9 nghmrcl1 24096 . . . . . . . . 9 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
10 nghmrcl2 24097 . . . . . . . . 9 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
117nmoge0 24085 . . . . . . . . 9 ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
129, 10, 1, 11syl3anc 1371 . . . . . . . 8 (𝐹 ∈ (𝑆 NGHom 𝑇) → 0 ≤ ((𝑆 normOp 𝑇)‘𝐹))
138, 12ge0p1rpd 12987 . . . . . . 7 (𝐹 ∈ (𝑆 NGHom 𝑇) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
1413adantr 481 . . . . . 6 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
156, 14rpdivcld 12974 . . . . 5 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) → (𝑟 / (((𝑆 normOp 𝑇)‘𝐹) + 1)) ∈ ℝ+)
16 ngpms 23956 . . . . . . . . . . . 12 (𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp)
179, 16syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ MetSp)
1817ad2antrr 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 2736 . . . . . . . . . . 11 (dist‘𝑆) = (dist‘𝑆)
222, 21mscl 23814 . . . . . . . . . 10 ((𝑆 ∈ MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
2318, 19, 20, 22syl3anc 1371 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝑥(dist‘𝑆)𝑦) ∈ ℝ)
246adantr 481 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ+)
2524rpred 12957 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑟 ∈ ℝ)
2613ad2antrr 724 . . . . . . . . 9 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ+)
2723, 25, 26ltmuldiv2d 13005 . . . . . . . 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 23956 . . . . . . . . . . . . 13 (𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp)
2910, 28syl 17 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ MetSp)
3029ad2antrr 724 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝑇 ∈ MetSp)
315ad2antrr 724 . . . . . . . . . . . 12 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 𝐹:(Base‘𝑆)⟶(Base‘𝑇))
3231, 19ffvelcdmd 7036 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹𝑥) ∈ (Base‘𝑇))
3331, 20ffvelcdmd 7036 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (𝐹𝑦) ∈ (Base‘𝑇))
34 eqid 2736 . . . . . . . . . . . 12 (dist‘𝑇) = (dist‘𝑇)
353, 34mscl 23814 . . . . . . . . . . 11 ((𝑇 ∈ MetSp ∧ (𝐹𝑥) ∈ (Base‘𝑇) ∧ (𝐹𝑦) ∈ (Base‘𝑇)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ∈ ℝ)
3630, 32, 33, 35syl3anc 1371 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ∈ ℝ)
378ad2antrr 724 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝑆 normOp 𝑇)‘𝐹) ∈ ℝ)
3837, 23remulcld 11185 . . . . . . . . . 10 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)) ∈ ℝ)
3926rpred 12957 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → (((𝑆 normOp 𝑇)‘𝐹) + 1) ∈ ℝ)
4039, 23remulcld 11185 . . . . . . . . . 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 24108 . . . . . . . . . . . 12 ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
42413expa 1118 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑥 ∈ (Base‘𝑆)) ∧ 𝑦 ∈ (Base‘𝑆)) → ((𝐹𝑥)(dist‘𝑇)(𝐹𝑦)) ≤ (((𝑆 normOp 𝑇)‘𝐹) · (𝑥(dist‘𝑆)𝑦)))
4342adantlrr 719 . . . . . . . . . 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 23807 . . . . . . . . . . . . 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 23816 . . . . . . . . . . . 12 ((𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ (Base‘𝑆) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
4745, 19, 20, 46syl3anc 1371 . . . . . . . . . . 11 (((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ (𝑥 ∈ (Base‘𝑆) ∧ 𝑟 ∈ ℝ+)) ∧ 𝑦 ∈ (Base‘𝑆)) → 0 ≤ (𝑥(dist‘𝑆)𝑦))
4837lep1d 12086 . . . . . . . . . . 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 12094 . . . . . . . . . 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 11312 . . . . . . . . 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 11245 . . . . . . . . . 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 1371 . . . . . . . . 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 693 . . . . . . . 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 259 . . . . . . 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 7521 . . . . . . . 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 5115 . . . . . . 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 7521 . . . . . . . 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 5115 . . . . . . 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 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)) → ((𝐹𝑥)((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))(𝐹𝑦)) < 𝑟))
6059ralrimiva 3143 . . . . 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 5109 . . . . . 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 3585 . . . . 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 3197 . . 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 2736 . . . . . 6 ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))) = ((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))
662, 65xmsxmet 23809 . . . . 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 23807 . . . . 5 (𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp)
69 eqid 2736 . . . . . 6 ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))) = ((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))
703, 69xmsxmet 23809 . . . . 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 2736 . . . . 5 (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆)))) = (MetOpen‘((dist‘𝑆) ↾ ((Base‘𝑆) × (Base‘𝑆))))
73 eqid 2736 . . . . 5 (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇)))) = (MetOpen‘((dist‘𝑇) ↾ ((Base‘𝑇) × (Base‘𝑇))))
7472, 73metcn 23899 . . . 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 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)
7877, 2, 65mstopn 23805 . . . 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 23805 . . . 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 7375 . 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 2841 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 3064  wrex 3073   class class class wbr 5105   × cxp 5631  cres 5635  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190   / cdiv 11812  +crp 12915  Basecbs 17083  distcds 17142  TopOpenctopn 17303   GrpHom cghm 19005  ∞Metcxmet 20781  MetOpencmopn 20786   Cn ccn 22575  ∞MetSpcxms 23670  MetSpcms 23671  NrmGrpcngp 23933   normOp cnmo 24069   NGHom cnghm 24070
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-0g 17323  df-topgen 17325  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-sbg 18753  df-ghm 19006  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cn 22578  df-cnp 22579  df-xms 23673  df-ms 23674  df-nm 23938  df-ngp 23939  df-nmo 24072  df-nghm 24073
This theorem is referenced by:  nmhmcn  24483
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