Step |
Hyp |
Ref |
Expression |
1 |
|
c0mhm.b |
|- B = ( Base ` S ) |
2 |
|
c0mhm.0 |
|- .0. = ( 0g ` T ) |
3 |
|
c0mhm.h |
|- H = ( x e. B |-> .0. ) |
4 |
|
eqid |
|- ( Base ` T ) = ( Base ` T ) |
5 |
4 2
|
mndidcl |
|- ( T e. Mnd -> .0. e. ( Base ` T ) ) |
6 |
5
|
adantl |
|- ( ( S e. Mnd /\ T e. Mnd ) -> .0. e. ( Base ` T ) ) |
7 |
6
|
adantr |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ x e. B ) -> .0. e. ( Base ` T ) ) |
8 |
7 3
|
fmptd |
|- ( ( S e. Mnd /\ T e. Mnd ) -> H : B --> ( Base ` T ) ) |
9 |
5
|
ancli |
|- ( T e. Mnd -> ( T e. Mnd /\ .0. e. ( Base ` T ) ) ) |
10 |
9
|
adantl |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( T e. Mnd /\ .0. e. ( Base ` T ) ) ) |
11 |
|
eqid |
|- ( +g ` T ) = ( +g ` T ) |
12 |
4 11 2
|
mndlid |
|- ( ( T e. Mnd /\ .0. e. ( Base ` T ) ) -> ( .0. ( +g ` T ) .0. ) = .0. ) |
13 |
10 12
|
syl |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( .0. ( +g ` T ) .0. ) = .0. ) |
14 |
13
|
adantr |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( .0. ( +g ` T ) .0. ) = .0. ) |
15 |
3
|
a1i |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> H = ( x e. B |-> .0. ) ) |
16 |
|
eqidd |
|- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = a ) -> .0. = .0. ) |
17 |
|
simprl |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> a e. B ) |
18 |
6
|
adantr |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> .0. e. ( Base ` T ) ) |
19 |
15 16 17 18
|
fvmptd |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` a ) = .0. ) |
20 |
|
eqidd |
|- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = b ) -> .0. = .0. ) |
21 |
|
simprr |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> b e. B ) |
22 |
15 20 21 18
|
fvmptd |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` b ) = .0. ) |
23 |
19 22
|
oveq12d |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( ( H ` a ) ( +g ` T ) ( H ` b ) ) = ( .0. ( +g ` T ) .0. ) ) |
24 |
|
eqidd |
|- ( ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) /\ x = ( a ( +g ` S ) b ) ) -> .0. = .0. ) |
25 |
|
eqid |
|- ( +g ` S ) = ( +g ` S ) |
26 |
1 25
|
mndcl |
|- ( ( S e. Mnd /\ a e. B /\ b e. B ) -> ( a ( +g ` S ) b ) e. B ) |
27 |
26
|
3expb |
|- ( ( S e. Mnd /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` S ) b ) e. B ) |
28 |
27
|
adantlr |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( a ( +g ` S ) b ) e. B ) |
29 |
15 24 28 18
|
fvmptd |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` ( a ( +g ` S ) b ) ) = .0. ) |
30 |
14 23 29
|
3eqtr4rd |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ ( a e. B /\ b e. B ) ) -> ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) ) |
31 |
30
|
ralrimivva |
|- ( ( S e. Mnd /\ T e. Mnd ) -> A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) ) |
32 |
3
|
a1i |
|- ( ( S e. Mnd /\ T e. Mnd ) -> H = ( x e. B |-> .0. ) ) |
33 |
|
eqidd |
|- ( ( ( S e. Mnd /\ T e. Mnd ) /\ x = ( 0g ` S ) ) -> .0. = .0. ) |
34 |
|
eqid |
|- ( 0g ` S ) = ( 0g ` S ) |
35 |
1 34
|
mndidcl |
|- ( S e. Mnd -> ( 0g ` S ) e. B ) |
36 |
35
|
adantr |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( 0g ` S ) e. B ) |
37 |
32 33 36 6
|
fvmptd |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( H ` ( 0g ` S ) ) = .0. ) |
38 |
8 31 37
|
3jca |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) ) |
39 |
38
|
ancli |
|- ( ( S e. Mnd /\ T e. Mnd ) -> ( ( S e. Mnd /\ T e. Mnd ) /\ ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) ) ) |
40 |
1 4 25 11 34 2
|
ismhm |
|- ( H e. ( S MndHom T ) <-> ( ( S e. Mnd /\ T e. Mnd ) /\ ( H : B --> ( Base ` T ) /\ A. a e. B A. b e. B ( H ` ( a ( +g ` S ) b ) ) = ( ( H ` a ) ( +g ` T ) ( H ` b ) ) /\ ( H ` ( 0g ` S ) ) = .0. ) ) ) |
41 |
39 40
|
sylibr |
|- ( ( S e. Mnd /\ T e. Mnd ) -> H e. ( S MndHom T ) ) |