Step |
Hyp |
Ref |
Expression |
1 |
|
nmf.x |
⊢ 𝑋 = ( Base ‘ 𝐺 ) |
2 |
|
nmf.n |
⊢ 𝑁 = ( norm ‘ 𝐺 ) |
3 |
|
nmeq0.z |
⊢ 0 = ( 0g ‘ 𝐺 ) |
4 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
5 |
2 1 3 4
|
nmval |
⊢ ( 𝐴 ∈ 𝑋 → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 ( dist ‘ 𝐺 ) 0 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( 𝑁 ‘ 𝐴 ) = ( 𝐴 ( dist ‘ 𝐺 ) 0 ) ) |
7 |
6
|
eqeq1d |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = 0 ↔ ( 𝐴 ( dist ‘ 𝐺 ) 0 ) = 0 ) ) |
8 |
|
ngpgrp |
⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ Grp ) |
9 |
8
|
adantr |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → 𝐺 ∈ Grp ) |
10 |
1 3
|
grpidcl |
⊢ ( 𝐺 ∈ Grp → 0 ∈ 𝑋 ) |
11 |
9 10
|
syl |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → 0 ∈ 𝑋 ) |
12 |
|
ngpxms |
⊢ ( 𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp ) |
13 |
1 4
|
xmseq0 |
⊢ ( ( 𝐺 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( ( 𝐴 ( dist ‘ 𝐺 ) 0 ) = 0 ↔ 𝐴 = 0 ) ) |
14 |
12 13
|
syl3an1 |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 0 ∈ 𝑋 ) → ( ( 𝐴 ( dist ‘ 𝐺 ) 0 ) = 0 ↔ 𝐴 = 0 ) ) |
15 |
11 14
|
mpd3an3 |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝐴 ( dist ‘ 𝐺 ) 0 ) = 0 ↔ 𝐴 = 0 ) ) |
16 |
7 15
|
bitrd |
⊢ ( ( 𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ) → ( ( 𝑁 ‘ 𝐴 ) = 0 ↔ 𝐴 = 0 ) ) |