Step |
Hyp |
Ref |
Expression |
1 |
|
zrrhm.b |
⊢ 𝐵 = ( Base ‘ 𝑇 ) |
2 |
|
zrrhm.0 |
⊢ 0 = ( 0g ‘ 𝑆 ) |
3 |
|
zrrhm.h |
⊢ 𝐻 = ( 𝑥 ∈ 𝐵 ↦ 0 ) |
4 |
|
c0snmhm.z |
⊢ 𝑍 = ( 0g ‘ 𝑇 ) |
5 |
|
pm3.22 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ) → ( 𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd ) ) |
6 |
5
|
3adant3 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → ( 𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd ) ) |
7 |
|
simp1 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝑆 ∈ Mnd ) |
8 |
|
mndmgm |
⊢ ( 𝑇 ∈ Mnd → 𝑇 ∈ Mgm ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝑇 ∈ Mgm ) |
10 |
|
fveq2 |
⊢ ( 𝐵 = { 𝑍 } → ( ♯ ‘ 𝐵 ) = ( ♯ ‘ { 𝑍 } ) ) |
11 |
4
|
fvexi |
⊢ 𝑍 ∈ V |
12 |
|
hashsng |
⊢ ( 𝑍 ∈ V → ( ♯ ‘ { 𝑍 } ) = 1 ) |
13 |
11 12
|
ax-mp |
⊢ ( ♯ ‘ { 𝑍 } ) = 1 |
14 |
10 13
|
eqtrdi |
⊢ ( 𝐵 = { 𝑍 } → ( ♯ ‘ 𝐵 ) = 1 ) |
15 |
14
|
3ad2ant3 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → ( ♯ ‘ 𝐵 ) = 1 ) |
16 |
1 2 3
|
c0snmgmhm |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ ( ♯ ‘ 𝐵 ) = 1 ) → 𝐻 ∈ ( 𝑇 MgmHom 𝑆 ) ) |
17 |
7 9 15 16
|
syl3anc |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝐻 ∈ ( 𝑇 MgmHom 𝑆 ) ) |
18 |
3
|
a1i |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝐻 = ( 𝑥 ∈ 𝐵 ↦ 0 ) ) |
19 |
|
eqidd |
⊢ ( ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) ∧ 𝑥 = 𝑍 ) → 0 = 0 ) |
20 |
11
|
snid |
⊢ 𝑍 ∈ { 𝑍 } |
21 |
|
eleq2 |
⊢ ( 𝐵 = { 𝑍 } → ( 𝑍 ∈ 𝐵 ↔ 𝑍 ∈ { 𝑍 } ) ) |
22 |
20 21
|
mpbiri |
⊢ ( 𝐵 = { 𝑍 } → 𝑍 ∈ 𝐵 ) |
23 |
22
|
3ad2ant3 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝑍 ∈ 𝐵 ) |
24 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
25 |
24 2
|
mndidcl |
⊢ ( 𝑆 ∈ Mnd → 0 ∈ ( Base ‘ 𝑆 ) ) |
26 |
25
|
3ad2ant1 |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 0 ∈ ( Base ‘ 𝑆 ) ) |
27 |
18 19 23 26
|
fvmptd |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → ( 𝐻 ‘ 𝑍 ) = 0 ) |
28 |
17 27
|
jca |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → ( 𝐻 ∈ ( 𝑇 MgmHom 𝑆 ) ∧ ( 𝐻 ‘ 𝑍 ) = 0 ) ) |
29 |
|
eqid |
⊢ ( +g ‘ 𝑇 ) = ( +g ‘ 𝑇 ) |
30 |
|
eqid |
⊢ ( +g ‘ 𝑆 ) = ( +g ‘ 𝑆 ) |
31 |
1 24 29 30 4 2
|
ismhm0 |
⊢ ( 𝐻 ∈ ( 𝑇 MndHom 𝑆 ) ↔ ( ( 𝑇 ∈ Mnd ∧ 𝑆 ∈ Mnd ) ∧ ( 𝐻 ∈ ( 𝑇 MgmHom 𝑆 ) ∧ ( 𝐻 ‘ 𝑍 ) = 0 ) ) ) |
32 |
6 28 31
|
sylanbrc |
⊢ ( ( 𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = { 𝑍 } ) → 𝐻 ∈ ( 𝑇 MndHom 𝑆 ) ) |