Step |
Hyp |
Ref |
Expression |
1 |
|
0nghm.2 |
⊢ 𝑉 = ( Base ‘ 𝑆 ) |
2 |
|
0nghm.3 |
⊢ 0 = ( 0g ‘ 𝑇 ) |
3 |
|
eqid |
⊢ ( 𝑆 normOp 𝑇 ) = ( 𝑆 normOp 𝑇 ) |
4 |
3 1 2
|
nmo0 |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( ( 𝑆 normOp 𝑇 ) ‘ ( 𝑉 × { 0 } ) ) = 0 ) |
5 |
|
0re |
⊢ 0 ∈ ℝ |
6 |
4 5
|
eqeltrdi |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( ( 𝑆 normOp 𝑇 ) ‘ ( 𝑉 × { 0 } ) ) ∈ ℝ ) |
7 |
|
ngpgrp |
⊢ ( 𝑆 ∈ NrmGrp → 𝑆 ∈ Grp ) |
8 |
|
ngpgrp |
⊢ ( 𝑇 ∈ NrmGrp → 𝑇 ∈ Grp ) |
9 |
2 1
|
0ghm |
⊢ ( ( 𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ) → ( 𝑉 × { 0 } ) ∈ ( 𝑆 GrpHom 𝑇 ) ) |
10 |
7 8 9
|
syl2an |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( 𝑉 × { 0 } ) ∈ ( 𝑆 GrpHom 𝑇 ) ) |
11 |
3
|
isnghm2 |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ ( 𝑉 × { 0 } ) ∈ ( 𝑆 GrpHom 𝑇 ) ) → ( ( 𝑉 × { 0 } ) ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( ( 𝑆 normOp 𝑇 ) ‘ ( 𝑉 × { 0 } ) ) ∈ ℝ ) ) |
12 |
10 11
|
mpd3an3 |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( ( 𝑉 × { 0 } ) ∈ ( 𝑆 NGHom 𝑇 ) ↔ ( ( 𝑆 normOp 𝑇 ) ‘ ( 𝑉 × { 0 } ) ) ∈ ℝ ) ) |
13 |
6 12
|
mpbird |
⊢ ( ( 𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ) → ( 𝑉 × { 0 } ) ∈ ( 𝑆 NGHom 𝑇 ) ) |