| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nmhmcn.j |
⊢ 𝐽 = ( TopOpen ‘ 𝑆 ) |
| 2 |
|
nmhmcn.k |
⊢ 𝐾 = ( TopOpen ‘ 𝑇 ) |
| 3 |
|
nmhmcn.g |
⊢ 𝐺 = ( Scalar ‘ 𝑆 ) |
| 4 |
|
nmhmcn.b |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
| 5 |
|
elinel1 |
⊢ ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) → 𝑆 ∈ NrmMod ) |
| 6 |
|
elinel1 |
⊢ ( 𝑇 ∈ ( NrmMod ∩ ℂMod ) → 𝑇 ∈ NrmMod ) |
| 7 |
|
isnmhm |
⊢ ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) ∧ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) ) |
| 8 |
7
|
baib |
⊢ ( ( 𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) ) |
| 9 |
5 6 8
|
syl2an |
⊢ ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) ) |
| 10 |
9
|
3adant3 |
⊢ ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) ) |
| 11 |
1 2
|
nghmcn |
⊢ ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
| 12 |
|
simpll1 |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑆 ∈ ( NrmMod ∩ ℂMod ) ) |
| 13 |
12
|
elin1d |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑆 ∈ NrmMod ) |
| 14 |
|
nlmngp |
⊢ ( 𝑆 ∈ NrmMod → 𝑆 ∈ NrmGrp ) |
| 15 |
|
ngpms |
⊢ ( 𝑆 ∈ NrmGrp → 𝑆 ∈ MetSp ) |
| 16 |
13 14 15
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑆 ∈ MetSp ) |
| 17 |
|
msxms |
⊢ ( 𝑆 ∈ MetSp → 𝑆 ∈ ∞MetSp ) |
| 18 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
| 19 |
|
eqid |
⊢ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) = ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) |
| 20 |
18 19
|
xmsxmet |
⊢ ( 𝑆 ∈ ∞MetSp → ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ) |
| 21 |
16 17 20
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ) |
| 22 |
|
simpr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) |
| 23 |
|
simpll2 |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑇 ∈ ( NrmMod ∩ ℂMod ) ) |
| 24 |
23
|
elin1d |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑇 ∈ NrmMod ) |
| 25 |
|
nlmngp |
⊢ ( 𝑇 ∈ NrmMod → 𝑇 ∈ NrmGrp ) |
| 26 |
|
ngpms |
⊢ ( 𝑇 ∈ NrmGrp → 𝑇 ∈ MetSp ) |
| 27 |
24 25 26
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑇 ∈ MetSp ) |
| 28 |
|
msxms |
⊢ ( 𝑇 ∈ MetSp → 𝑇 ∈ ∞MetSp ) |
| 29 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
| 30 |
|
eqid |
⊢ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) = ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) |
| 31 |
29 30
|
xmsxmet |
⊢ ( 𝑇 ∈ ∞MetSp → ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ) |
| 32 |
27 28 31
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ) |
| 33 |
|
nlmlmod |
⊢ ( 𝑇 ∈ NrmMod → 𝑇 ∈ LMod ) |
| 34 |
|
eqid |
⊢ ( 0g ‘ 𝑇 ) = ( 0g ‘ 𝑇 ) |
| 35 |
29 34
|
lmod0vcl |
⊢ ( 𝑇 ∈ LMod → ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ) |
| 36 |
24 33 35
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ) |
| 37 |
|
1rp |
⊢ 1 ∈ ℝ+ |
| 38 |
|
rpxr |
⊢ ( 1 ∈ ℝ+ → 1 ∈ ℝ* ) |
| 39 |
37 38
|
mp1i |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 1 ∈ ℝ* ) |
| 40 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) |
| 41 |
40
|
blopn |
⊢ ( ( ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ∧ ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ∧ 1 ∈ ℝ* ) → ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ∈ ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 42 |
32 36 39 41
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ∈ ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 43 |
2 29 30
|
mstopn |
⊢ ( 𝑇 ∈ MetSp → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 44 |
24 25 26 43
|
4syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐾 = ( MetOpen ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) ) |
| 45 |
42 44
|
eleqtrrd |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ∈ 𝐾 ) |
| 46 |
|
cnima |
⊢ ( ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ∧ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ∈ 𝐾 ) → ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ∈ 𝐽 ) |
| 47 |
22 45 46
|
syl2anc |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ∈ 𝐽 ) |
| 48 |
1 18 19
|
mstopn |
⊢ ( 𝑆 ∈ MetSp → 𝐽 = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ) |
| 49 |
13 14 15 48
|
4syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐽 = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ) |
| 50 |
47 49
|
eleqtrd |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ∈ ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ) |
| 51 |
|
nlmlmod |
⊢ ( 𝑆 ∈ NrmMod → 𝑆 ∈ LMod ) |
| 52 |
|
eqid |
⊢ ( 0g ‘ 𝑆 ) = ( 0g ‘ 𝑆 ) |
| 53 |
18 52
|
lmod0vcl |
⊢ ( 𝑆 ∈ LMod → ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ) |
| 54 |
13 51 53
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ) |
| 55 |
|
lmghm |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
| 56 |
55
|
ad2antlr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) |
| 57 |
52 34
|
ghmid |
⊢ ( 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) |
| 58 |
56 57
|
syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) = ( 0g ‘ 𝑇 ) ) |
| 59 |
37
|
a1i |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 1 ∈ ℝ+ ) |
| 60 |
|
blcntr |
⊢ ( ( ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ∧ ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ∧ 1 ∈ ℝ+ ) → ( 0g ‘ 𝑇 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) |
| 61 |
32 36 59 60
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 0g ‘ 𝑇 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) |
| 62 |
58 61
|
eqeltrd |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) |
| 63 |
18 29
|
lmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 64 |
63
|
ad2antlr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 65 |
|
ffn |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → 𝐹 Fn ( Base ‘ 𝑆 ) ) |
| 66 |
|
elpreima |
⊢ ( 𝐹 Fn ( Base ‘ 𝑆 ) → ( ( 0g ‘ 𝑆 ) ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ( ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 67 |
64 65 66
|
3syl |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ( 0g ‘ 𝑆 ) ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ( ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ ( 0g ‘ 𝑆 ) ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 68 |
54 62 67
|
mpbir2and |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 0g ‘ 𝑆 ) ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) |
| 69 |
|
eqid |
⊢ ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) = ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) |
| 70 |
69
|
mopni2 |
⊢ ( ( ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ∧ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ∈ ( MetOpen ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) ∧ ( 0g ‘ 𝑆 ) ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) → ∃ 𝑥 ∈ ℝ+ ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) |
| 71 |
21 50 68 70
|
syl3anc |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ∃ 𝑥 ∈ ℝ+ ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) |
| 72 |
|
simpl1 |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑆 ∈ ( NrmMod ∩ ℂMod ) ) |
| 73 |
72
|
elin1d |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑆 ∈ NrmMod ) |
| 74 |
73 14
|
syl |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑆 ∈ NrmGrp ) |
| 75 |
74
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑆 ∈ NrmGrp ) |
| 76 |
75
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑆 ∈ NrmGrp ) |
| 77 |
|
ngpgrp |
⊢ ( 𝑆 ∈ NrmGrp → 𝑆 ∈ Grp ) |
| 78 |
76 77
|
syl |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑆 ∈ Grp ) |
| 79 |
|
simpr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑦 ∈ ( Base ‘ 𝑆 ) ) |
| 80 |
|
eqid |
⊢ ( norm ‘ 𝑆 ) = ( norm ‘ 𝑆 ) |
| 81 |
|
eqid |
⊢ ( dist ‘ 𝑆 ) = ( dist ‘ 𝑆 ) |
| 82 |
80 18 52 81 19
|
nmval2 |
⊢ ( ( 𝑆 ∈ Grp ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) = ( 𝑦 ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ( 0g ‘ 𝑆 ) ) ) |
| 83 |
78 79 82
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) = ( 𝑦 ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ( 0g ‘ 𝑆 ) ) ) |
| 84 |
21
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ) |
| 85 |
54
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ) |
| 86 |
|
xmetsym |
⊢ ( ( ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ) → ( 𝑦 ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ( 0g ‘ 𝑆 ) ) = ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) ) |
| 87 |
84 79 85 86
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑦 ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ( 0g ‘ 𝑆 ) ) = ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) ) |
| 88 |
83 87
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) = ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) ) |
| 89 |
88
|
breq1d |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) < 𝑥 ↔ ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 ) ) |
| 90 |
89
|
biimpd |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) < 𝑥 → ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 ) ) |
| 91 |
64
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 92 |
|
elpreima |
⊢ ( 𝐹 Fn ( Base ‘ 𝑆 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 93 |
91 65 92
|
3syl |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 94 |
32
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ) |
| 95 |
36
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ) |
| 96 |
37 38
|
mp1i |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 1 ∈ ℝ* ) |
| 97 |
|
elbl |
⊢ ( ( ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ∧ ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ∧ 1 ∈ ℝ* ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ↔ ( ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ∧ ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) < 1 ) ) ) |
| 98 |
94 95 96 97
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ↔ ( ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ∧ ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) < 1 ) ) ) |
| 99 |
|
simpl2 |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑇 ∈ ( NrmMod ∩ ℂMod ) ) |
| 100 |
99
|
elin1d |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑇 ∈ NrmMod ) |
| 101 |
100 25
|
syl |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → 𝑇 ∈ NrmGrp ) |
| 102 |
101
|
adantr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝑇 ∈ NrmGrp ) |
| 103 |
102
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑇 ∈ NrmGrp ) |
| 104 |
|
simplr |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
| 105 |
104
|
adantr |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
| 106 |
105 63
|
syl |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
| 107 |
106
|
ffvelrnda |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ) |
| 108 |
|
eqid |
⊢ ( norm ‘ 𝑇 ) = ( norm ‘ 𝑇 ) |
| 109 |
29 108
|
nmcl |
⊢ ( ( 𝑇 ∈ NrmGrp ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ∈ ℝ ) |
| 110 |
103 107 109
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ∈ ℝ ) |
| 111 |
|
1re |
⊢ 1 ∈ ℝ |
| 112 |
|
ltle |
⊢ ( ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) < 1 → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ) ) |
| 113 |
110 111 112
|
sylancl |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) < 1 → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ) ) |
| 114 |
|
ngpgrp |
⊢ ( 𝑇 ∈ NrmGrp → 𝑇 ∈ Grp ) |
| 115 |
103 114
|
syl |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑇 ∈ Grp ) |
| 116 |
|
eqid |
⊢ ( dist ‘ 𝑇 ) = ( dist ‘ 𝑇 ) |
| 117 |
108 29 34 116 30
|
nmval2 |
⊢ ( ( 𝑇 ∈ Grp ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 0g ‘ 𝑇 ) ) ) |
| 118 |
115 107 117
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) = ( ( 𝐹 ‘ 𝑦 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 0g ‘ 𝑇 ) ) ) |
| 119 |
|
xmetsym |
⊢ ( ( ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑇 ) ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ∧ ( 0g ‘ 𝑇 ) ∈ ( Base ‘ 𝑇 ) ) → ( ( 𝐹 ‘ 𝑦 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 0g ‘ 𝑇 ) ) = ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
| 120 |
94 107 95 119
|
syl3anc |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑦 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 0g ‘ 𝑇 ) ) = ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
| 121 |
118 120
|
eqtrd |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) = ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) ) |
| 122 |
121
|
breq1d |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) < 1 ↔ ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) < 1 ) ) |
| 123 |
|
1red |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 1 ∈ ℝ ) |
| 124 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → 𝑥 ∈ ℝ+ ) |
| 125 |
110 123 124
|
lediv1d |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) ≤ 1 ↔ ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 126 |
113 122 125
|
3imtr3d |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) < 1 → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 127 |
126
|
adantld |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( 𝐹 ‘ 𝑦 ) ∈ ( Base ‘ 𝑇 ) ∧ ( ( 0g ‘ 𝑇 ) ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ( 𝐹 ‘ 𝑦 ) ) < 1 ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 128 |
98 127
|
sylbid |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 129 |
128
|
adantld |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( 𝑦 ∈ ( Base ‘ 𝑆 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 130 |
93 129
|
sylbid |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) |
| 131 |
90 130
|
imim12d |
⊢ ( ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) ∧ 𝑦 ∈ ( Base ‘ 𝑆 ) ) → ( ( ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 → 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) → ( ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) < 𝑥 → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) ) |
| 132 |
131
|
ralimdva |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 → 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) → ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) < 𝑥 → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) ) |
| 133 |
|
rpxr |
⊢ ( 𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ* ) |
| 134 |
|
blval |
⊢ ( ( ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ∈ ( ∞Met ‘ ( Base ‘ 𝑆 ) ) ∧ ( 0g ‘ 𝑆 ) ∈ ( Base ‘ 𝑆 ) ∧ 𝑥 ∈ ℝ* ) → ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) = { 𝑦 ∈ ( Base ‘ 𝑆 ) ∣ ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 } ) |
| 135 |
21 54 133 134
|
syl2an3an |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) = { 𝑦 ∈ ( Base ‘ 𝑆 ) ∣ ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 } ) |
| 136 |
135
|
sseq1d |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ { 𝑦 ∈ ( Base ‘ 𝑆 ) ∣ ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 } ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 137 |
|
rabss |
⊢ ( { 𝑦 ∈ ( Base ‘ 𝑆 ) ∣ ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 } ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 → 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) |
| 138 |
136 137
|
bitrdi |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( 0g ‘ 𝑆 ) ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) 𝑦 ) < 𝑥 → 𝑦 ∈ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) ) ) ) |
| 139 |
|
eqid |
⊢ ( 𝑆 normOp 𝑇 ) = ( 𝑆 normOp 𝑇 ) |
| 140 |
12
|
adantr |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝑆 ∈ ( NrmMod ∩ ℂMod ) ) |
| 141 |
23
|
adantr |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝑇 ∈ ( NrmMod ∩ ℂMod ) ) |
| 142 |
|
rpreccl |
⊢ ( 𝑥 ∈ ℝ+ → ( 1 / 𝑥 ) ∈ ℝ+ ) |
| 143 |
142
|
adantl |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( 1 / 𝑥 ) ∈ ℝ+ ) |
| 144 |
143
|
rpxrd |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( 1 / 𝑥 ) ∈ ℝ* ) |
| 145 |
|
simpr |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → 𝑥 ∈ ℝ+ ) |
| 146 |
|
simpl3 |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ℚ ⊆ 𝐵 ) |
| 147 |
146
|
ad2antrr |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ℚ ⊆ 𝐵 ) |
| 148 |
139 18 80 108 3 4 140 141 105 144 145 147
|
nmoleub2b |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( 1 / 𝑥 ) ↔ ∀ 𝑦 ∈ ( Base ‘ 𝑆 ) ( ( ( norm ‘ 𝑆 ) ‘ 𝑦 ) < 𝑥 → ( ( ( norm ‘ 𝑇 ) ‘ ( 𝐹 ‘ 𝑦 ) ) / 𝑥 ) ≤ ( 1 / 𝑥 ) ) ) ) |
| 149 |
132 138 148
|
3imtr4d |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) → ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( 1 / 𝑥 ) ) ) |
| 150 |
75 102 56
|
3jca |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) ) |
| 151 |
142
|
rpred |
⊢ ( 𝑥 ∈ ℝ+ → ( 1 / 𝑥 ) ∈ ℝ ) |
| 152 |
139
|
bddnghm |
⊢ ( ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) ∧ ( ( 1 / 𝑥 ) ∈ ℝ ∧ ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( 1 / 𝑥 ) ) ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) |
| 153 |
152
|
expr |
⊢ ( ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ ( 𝑆 GrpHom 𝑇 ) ) ∧ ( 1 / 𝑥 ) ∈ ℝ ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( 1 / 𝑥 ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) |
| 154 |
150 151 153
|
syl2an |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 𝑆 normOp 𝑇 ) ‘ 𝐹 ) ≤ ( 1 / 𝑥 ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) |
| 155 |
149 154
|
syld |
⊢ ( ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ∧ 𝑥 ∈ ℝ+ ) → ( ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) |
| 156 |
155
|
rexlimdva |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → ( ∃ 𝑥 ∈ ℝ+ ( ( 0g ‘ 𝑆 ) ( ball ‘ ( ( dist ‘ 𝑆 ) ↾ ( ( Base ‘ 𝑆 ) × ( Base ‘ 𝑆 ) ) ) ) 𝑥 ) ⊆ ( ◡ 𝐹 “ ( ( 0g ‘ 𝑇 ) ( ball ‘ ( ( dist ‘ 𝑇 ) ↾ ( ( Base ‘ 𝑇 ) × ( Base ‘ 𝑇 ) ) ) ) 1 ) ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) |
| 157 |
71 156
|
mpd |
⊢ ( ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) |
| 158 |
157
|
ex |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) → 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ) |
| 159 |
11 158
|
impbid2 |
⊢ ( ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) ∧ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) → ( 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ↔ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ) |
| 160 |
159
|
pm5.32da |
⊢ ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) → ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝑆 NGHom 𝑇 ) ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ) ) |
| 161 |
10 160
|
bitrd |
⊢ ( ( 𝑆 ∈ ( NrmMod ∩ ℂMod ) ∧ 𝑇 ∈ ( NrmMod ∩ ℂMod ) ∧ ℚ ⊆ 𝐵 ) → ( 𝐹 ∈ ( 𝑆 NMHom 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ) ) ) |