Step |
Hyp |
Ref |
Expression |
1 |
|
0lmhm.z |
⊢ 0 = ( 0g ‘ 𝑁 ) |
2 |
|
0lmhm.b |
⊢ 𝐵 = ( Base ‘ 𝑀 ) |
3 |
|
0lmhm.s |
⊢ 𝑆 = ( Scalar ‘ 𝑀 ) |
4 |
|
0lmhm.t |
⊢ 𝑇 = ( Scalar ‘ 𝑁 ) |
5 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑀 ) = ( ·𝑠 ‘ 𝑀 ) |
6 |
|
eqid |
⊢ ( ·𝑠 ‘ 𝑁 ) = ( ·𝑠 ‘ 𝑁 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
8 |
|
simp1 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → 𝑀 ∈ LMod ) |
9 |
|
simp2 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → 𝑁 ∈ LMod ) |
10 |
|
simp3 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → 𝑆 = 𝑇 ) |
11 |
10
|
eqcomd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → 𝑇 = 𝑆 ) |
12 |
|
lmodgrp |
⊢ ( 𝑀 ∈ LMod → 𝑀 ∈ Grp ) |
13 |
|
lmodgrp |
⊢ ( 𝑁 ∈ LMod → 𝑁 ∈ Grp ) |
14 |
1 2
|
0ghm |
⊢ ( ( 𝑀 ∈ Grp ∧ 𝑁 ∈ Grp ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 GrpHom 𝑁 ) ) |
15 |
12 13 14
|
syl2an |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 GrpHom 𝑁 ) ) |
16 |
15
|
3adant3 |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 GrpHom 𝑁 ) ) |
17 |
|
simpl2 |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑁 ∈ LMod ) |
18 |
|
simprl |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ ( Base ‘ 𝑆 ) ) |
19 |
|
simpl3 |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑆 = 𝑇 ) |
20 |
19
|
fveq2d |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( Base ‘ 𝑆 ) = ( Base ‘ 𝑇 ) ) |
21 |
18 20
|
eleqtrd |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑥 ∈ ( Base ‘ 𝑇 ) ) |
22 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
23 |
4 6 22 1
|
lmodvs0 |
⊢ ( ( 𝑁 ∈ LMod ∧ 𝑥 ∈ ( Base ‘ 𝑇 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑁 ) 0 ) = 0 ) |
24 |
17 21 23
|
syl2anc |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑁 ) 0 ) = 0 ) |
25 |
1
|
fvexi |
⊢ 0 ∈ V |
26 |
25
|
fvconst2 |
⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) = 0 ) |
27 |
26
|
oveq2d |
⊢ ( 𝑦 ∈ 𝐵 → ( 𝑥 ( ·𝑠 ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑁 ) 0 ) ) |
28 |
27
|
ad2antll |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑁 ) 0 ) ) |
29 |
|
simpl1 |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑀 ∈ LMod ) |
30 |
|
simprr |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → 𝑦 ∈ 𝐵 ) |
31 |
2 3 5 7
|
lmodvscl |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
32 |
29 18 30 31
|
syl3anc |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 ) |
33 |
25
|
fvconst2 |
⊢ ( ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ∈ 𝐵 → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) = 0 ) |
34 |
32 33
|
syl |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) = 0 ) |
35 |
24 28 34
|
3eqtr4rd |
⊢ ( ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) ∧ ( 𝑥 ∈ ( Base ‘ 𝑆 ) ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝐵 × { 0 } ) ‘ ( 𝑥 ( ·𝑠 ‘ 𝑀 ) 𝑦 ) ) = ( 𝑥 ( ·𝑠 ‘ 𝑁 ) ( ( 𝐵 × { 0 } ) ‘ 𝑦 ) ) ) |
36 |
2 5 6 3 4 7 8 9 11 16 35
|
islmhmd |
⊢ ( ( 𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇 ) → ( 𝐵 × { 0 } ) ∈ ( 𝑀 LMHom 𝑁 ) ) |