Step |
Hyp |
Ref |
Expression |
1 |
|
imassrn |
|- ( F " X ) C_ ran F |
2 |
|
eqid |
|- ( Base ` M ) = ( Base ` M ) |
3 |
|
eqid |
|- ( Base ` N ) = ( Base ` N ) |
4 |
2 3
|
mhmf |
|- ( F e. ( M MndHom N ) -> F : ( Base ` M ) --> ( Base ` N ) ) |
5 |
4
|
adantr |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> F : ( Base ` M ) --> ( Base ` N ) ) |
6 |
5
|
frnd |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ran F C_ ( Base ` N ) ) |
7 |
1 6
|
sstrid |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( F " X ) C_ ( Base ` N ) ) |
8 |
|
eqid |
|- ( 0g ` M ) = ( 0g ` M ) |
9 |
|
eqid |
|- ( 0g ` N ) = ( 0g ` N ) |
10 |
8 9
|
mhm0 |
|- ( F e. ( M MndHom N ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) ) |
11 |
10
|
adantr |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) = ( 0g ` N ) ) |
12 |
5
|
ffnd |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> F Fn ( Base ` M ) ) |
13 |
2
|
submss |
|- ( X e. ( SubMnd ` M ) -> X C_ ( Base ` M ) ) |
14 |
13
|
adantl |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> X C_ ( Base ` M ) ) |
15 |
8
|
subm0cl |
|- ( X e. ( SubMnd ` M ) -> ( 0g ` M ) e. X ) |
16 |
15
|
adantl |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( 0g ` M ) e. X ) |
17 |
|
fnfvima |
|- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( 0g ` M ) e. X ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) ) |
18 |
12 14 16 17
|
syl3anc |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( F ` ( 0g ` M ) ) e. ( F " X ) ) |
19 |
11 18
|
eqeltrrd |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( 0g ` N ) e. ( F " X ) ) |
20 |
|
simpll |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F e. ( M MndHom N ) ) |
21 |
14
|
adantr |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> X C_ ( Base ` M ) ) |
22 |
|
simprl |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. X ) |
23 |
21 22
|
sseldd |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> z e. ( Base ` M ) ) |
24 |
|
simprr |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. X ) |
25 |
21 24
|
sseldd |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> x e. ( Base ` M ) ) |
26 |
|
eqid |
|- ( +g ` M ) = ( +g ` M ) |
27 |
|
eqid |
|- ( +g ` N ) = ( +g ` N ) |
28 |
2 26 27
|
mhmlin |
|- ( ( F e. ( M MndHom N ) /\ z e. ( Base ` M ) /\ x e. ( Base ` M ) ) -> ( F ` ( z ( +g ` M ) x ) ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) ) |
29 |
20 23 25 28
|
syl3anc |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( F ` ( z ( +g ` M ) x ) ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) ) |
30 |
12
|
adantr |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> F Fn ( Base ` M ) ) |
31 |
26
|
submcl |
|- ( ( X e. ( SubMnd ` M ) /\ z e. X /\ x e. X ) -> ( z ( +g ` M ) x ) e. X ) |
32 |
31
|
3expb |
|- ( ( X e. ( SubMnd ` M ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X ) |
33 |
32
|
adantll |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( z ( +g ` M ) x ) e. X ) |
34 |
|
fnfvima |
|- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) /\ ( z ( +g ` M ) x ) e. X ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) ) |
35 |
30 21 33 34
|
syl3anc |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( F ` ( z ( +g ` M ) x ) ) e. ( F " X ) ) |
36 |
29 35
|
eqeltrrd |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ ( z e. X /\ x e. X ) ) -> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) |
37 |
36
|
anassrs |
|- ( ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ z e. X ) /\ x e. X ) -> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) |
38 |
37
|
ralrimiva |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ z e. X ) -> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) |
39 |
|
oveq2 |
|- ( y = ( F ` x ) -> ( ( F ` z ) ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) ( F ` x ) ) ) |
40 |
39
|
eleq1d |
|- ( y = ( F ` x ) -> ( ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) ) |
41 |
40
|
ralima |
|- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) ) |
42 |
12 14 41
|
syl2anc |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) ) |
43 |
42
|
adantr |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ z e. X ) -> ( A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) <-> A. x e. X ( ( F ` z ) ( +g ` N ) ( F ` x ) ) e. ( F " X ) ) ) |
44 |
38 43
|
mpbird |
|- ( ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) /\ z e. X ) -> A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) |
45 |
44
|
ralrimiva |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) |
46 |
|
oveq1 |
|- ( x = ( F ` z ) -> ( x ( +g ` N ) y ) = ( ( F ` z ) ( +g ` N ) y ) ) |
47 |
46
|
eleq1d |
|- ( x = ( F ` z ) -> ( ( x ( +g ` N ) y ) e. ( F " X ) <-> ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) ) |
48 |
47
|
ralbidv |
|- ( x = ( F ` z ) -> ( A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) ) |
49 |
48
|
ralima |
|- ( ( F Fn ( Base ` M ) /\ X C_ ( Base ` M ) ) -> ( A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) ) |
50 |
12 14 49
|
syl2anc |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) <-> A. z e. X A. y e. ( F " X ) ( ( F ` z ) ( +g ` N ) y ) e. ( F " X ) ) ) |
51 |
45 50
|
mpbird |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) ) |
52 |
|
mhmrcl2 |
|- ( F e. ( M MndHom N ) -> N e. Mnd ) |
53 |
52
|
adantr |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> N e. Mnd ) |
54 |
3 9 27
|
issubm |
|- ( N e. Mnd -> ( ( F " X ) e. ( SubMnd ` N ) <-> ( ( F " X ) C_ ( Base ` N ) /\ ( 0g ` N ) e. ( F " X ) /\ A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) ) ) ) |
55 |
53 54
|
syl |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( ( F " X ) e. ( SubMnd ` N ) <-> ( ( F " X ) C_ ( Base ` N ) /\ ( 0g ` N ) e. ( F " X ) /\ A. x e. ( F " X ) A. y e. ( F " X ) ( x ( +g ` N ) y ) e. ( F " X ) ) ) ) |
56 |
7 19 51 55
|
mpbir3and |
|- ( ( F e. ( M MndHom N ) /\ X e. ( SubMnd ` M ) ) -> ( F " X ) e. ( SubMnd ` N ) ) |