| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nghmcn.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑆 ) |
| 2 |
|
nghmcn.k |
⊢ 𝐾 = ( TopOpen ‘ 𝑇 ) |
| 3 |
|
nghmghm |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
| 4 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
| 5 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
| 6 |
4 5
|
ghmf |
⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 7 |
3 6
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 8 |
|
simprr |
⊢ ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) → 𝑟 ∈ ℝ+ ) |
| 9 |
|
eqid |
⊢ ( 𝑆 normOp 𝑇 ) = ( 𝑆 normOp 𝑇 ) |
| 10 |
9
|
nghmcl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ∈ ℝ ) |
| 11 |
|
nghmrcl1 |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑆 ∈ NrmGrp ) |
| 12 |
|
nghmrcl2 |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑇 ∈ NrmGrp ) |
| 13 |
9
|
nmoge0 |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) → 0 ≤ ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ) |
| 14 |
11 12 3 13
|
syl3anc |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 0 ≤ ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ) |
| 15 |
10 14
|
ge0p1rpd |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ∈ ℝ+ ) |
| 16 |
15
|
adantr |
⊢ ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ∈ ℝ+ ) |
| 17 |
8 16
|
rpdivcld |
⊢ ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) → ( 𝑟 / ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ) ∈ ℝ+ ) |
| 18 |
|
ngpms |
⊢ ( 𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp ) |
| 19 |
11 18
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑆 ∈ MetSp ) |
| 20 |
19
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑆 ∈ MetSp ) |
| 21 |
|
simplrl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) ) |
| 22 |
|
simpr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) ) |
| 23 |
|
eqid |
⊢ ( dist ‘ 𝑆 ) = ( dist ‘ 𝑆 ) |
| 24 |
4 23
|
mscl |
⊢ ( ( 𝑆 ∈ MetSp ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ∈ ℝ ) |
| 25 |
20 21 22 24
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ∈ ℝ ) |
| 26 |
8
|
adantr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑟 ∈ ℝ+ ) |
| 27 |
26
|
rpred |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑟 ∈ ℝ ) |
| 28 |
15
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ∈ ℝ+ ) |
| 29 |
25 27 28
|
ltmuldiv2d |
⊢ ( ( ( 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 ) ) ) ) |
| 30 |
|
ngpms |
⊢ ( 𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp ) |
| 31 |
12 30
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝑇 ∈ MetSp ) |
| 32 |
31
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑇 ∈ MetSp ) |
| 33 |
7
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 34 |
33 21
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ) |
| 35 |
33 22
|
ffvelrnd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ) |
| 36 |
|
eqid |
⊢ ( dist ‘ 𝑇 ) = ( dist ‘ 𝑇 ) |
| 37 |
5 36
|
mscl |
⊢ ( ( 𝑇 ∈ MetSp ∧ ( 𝐹 ‘ 𝑥 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ℝ ) |
| 38 |
32 34 35 37
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ∈ ℝ ) |
| 39 |
10
|
ad2antrr |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ∈ ℝ ) |
| 40 |
39 25
|
remulcld |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) · ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) ∈ ℝ ) |
| 41 |
28
|
rpred |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ∈ ℝ ) |
| 42 |
41 25
|
remulcld |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) · ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) ∈ ℝ ) |
| 43 |
9 4 23 36
|
nmods |
⊢ ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ≤ ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) · ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) ) |
| 44 |
43
|
3expa |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑥 ) ( dist ‘ 𝑇 ) ( 𝐹 ‘ 𝑦 ) ) ≤ ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) · ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) ) |
| 45 |
44
|
adantlrr |
⊢ ( ( ( 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 ) ) ) |
| 46 |
|
msxms |
⊢ ( 𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp ) |
| 47 |
20 46
|
syl |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑆 ∈ ∞MetSp ) |
| 48 |
4 23
|
xmsge0 |
⊢ ( ( 𝑆 ∈ ∞MetSp ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 0 ≤ ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) |
| 49 |
47 21 22 48
|
syl3anc |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 0 ≤ ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) |
| 50 |
39
|
lep1d |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) + 1 ) ) |
| 51 |
39 41 25 49 50
|
lemul1ad |
⊢ ( ( ( 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 ) ) ) |
| 52 |
38 40 42 45 51
|
letrd |
⊢ ( ( ( 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 ) ) ) |
| 53 |
|
lelttr |
⊢ ( ( ( ( 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 ) ) |
| 54 |
38 42 27 53
|
syl3anc |
⊢ ( ( ( 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 ) ) |
| 55 |
52 54
|
mpand |
⊢ ( ( ( 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 ) ) |
| 56 |
29 55
|
sylbird |
⊢ ( ( ( 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 ) ) |
| 57 |
21 22
|
ovresd |
⊢ ( ( ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑟 ∈ ℝ+ ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑥 ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) = ( 𝑥 ( dist ‘ 𝑆 ) 𝑦 ) ) |
| 58 |
57
|
breq1d |
⊢ ( ( ( 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 ) ) ) ) |
| 59 |
34 35
|
ovresd |
⊢ ( ( ( 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 ) ) ) |
| 60 |
59
|
breq1d |
⊢ ( ( ( 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 ) ) |
| 61 |
56 58 60
|
3imtr4d |
⊢ ( ( ( 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 ) ) |
| 62 |
61
|
ralrimiva |
⊢ ( ( 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 ) ) |
| 63 |
|
breq2 |
⊢ ( 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 ) ) ) ) |
| 64 |
63
|
rspceaimv |
⊢ ( ( ( 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 |
17 62 64
|
syl2anc |
⊢ ( ( 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 |
⊢ ( 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 |
⊢ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) = ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) |
| 68 |
4 67
|
xmsxmet |
⊢ ( 𝑆 ∈ ∞MetSp → ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ) |
| 69 |
19 46 68
|
3syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ) |
| 70 |
|
msxms |
⊢ ( 𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp ) |
| 71 |
|
eqid |
⊢ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) = ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) |
| 72 |
5 71
|
xmsxmet |
⊢ ( 𝑇 ∈ ∞MetSp → ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ) |
| 73 |
31 70 72
|
3syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ) |
| 74 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) |
| 75 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) |
| 76 |
74 75
|
metcn |
⊢ ( ( ( ( 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 |
⊢ ( 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 |
7 66 77
|
mpbir2and |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐹 ∈ ( ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) ) |
| 79 |
1 4 67
|
mstopn |
⊢ ( 𝑆 ∈ MetSp → 𝐽 = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ) |
| 80 |
19 79
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐽 = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ) |
| 81 |
2 5 71
|
mstopn |
⊢ ( 𝑇 ∈ MetSp → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 82 |
31 81
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 83 |
80 82
|
oveq12d |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → ( 𝐽 Cn 𝐾 ) = ( ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) Cn ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) ) |
| 84 |
78 83
|
eleqtrrd |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |