Step |
Hyp |
Ref |
Expression |
1 |
|
imasmnd.u |
|- ( ph -> U = ( F "s R ) ) |
2 |
|
imasmnd.v |
|- ( ph -> V = ( Base ` R ) ) |
3 |
|
imasmnd.p |
|- .+ = ( +g ` R ) |
4 |
|
imasmnd.f |
|- ( ph -> F : V -onto-> B ) |
5 |
|
imasmnd.e |
|- ( ( ph /\ ( a e. V /\ b e. V ) /\ ( p e. V /\ q e. V ) ) -> ( ( ( F ` a ) = ( F ` p ) /\ ( F ` b ) = ( F ` q ) ) -> ( F ` ( a .+ b ) ) = ( F ` ( p .+ q ) ) ) ) |
6 |
|
imasmnd.r |
|- ( ph -> R e. Mnd ) |
7 |
|
imasmnd.z |
|- .0. = ( 0g ` R ) |
8 |
6
|
3ad2ant1 |
|- ( ( ph /\ x e. V /\ y e. V ) -> R e. Mnd ) |
9 |
|
simp2 |
|- ( ( ph /\ x e. V /\ y e. V ) -> x e. V ) |
10 |
2
|
3ad2ant1 |
|- ( ( ph /\ x e. V /\ y e. V ) -> V = ( Base ` R ) ) |
11 |
9 10
|
eleqtrd |
|- ( ( ph /\ x e. V /\ y e. V ) -> x e. ( Base ` R ) ) |
12 |
|
simp3 |
|- ( ( ph /\ x e. V /\ y e. V ) -> y e. V ) |
13 |
12 10
|
eleqtrd |
|- ( ( ph /\ x e. V /\ y e. V ) -> y e. ( Base ` R ) ) |
14 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
15 |
14 3
|
mndcl |
|- ( ( R e. Mnd /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x .+ y ) e. ( Base ` R ) ) |
16 |
8 11 13 15
|
syl3anc |
|- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. ( Base ` R ) ) |
17 |
16 10
|
eleqtrrd |
|- ( ( ph /\ x e. V /\ y e. V ) -> ( x .+ y ) e. V ) |
18 |
6
|
adantr |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> R e. Mnd ) |
19 |
11
|
3adant3r3 |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> x e. ( Base ` R ) ) |
20 |
13
|
3adant3r3 |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> y e. ( Base ` R ) ) |
21 |
|
simpr3 |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. V ) |
22 |
2
|
adantr |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> V = ( Base ` R ) ) |
23 |
21 22
|
eleqtrd |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> z e. ( Base ` R ) ) |
24 |
14 3
|
mndass |
|- ( ( R e. Mnd /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) |
25 |
18 19 20 23 24
|
syl13anc |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( ( x .+ y ) .+ z ) = ( x .+ ( y .+ z ) ) ) |
26 |
25
|
fveq2d |
|- ( ( ph /\ ( x e. V /\ y e. V /\ z e. V ) ) -> ( F ` ( ( x .+ y ) .+ z ) ) = ( F ` ( x .+ ( y .+ z ) ) ) ) |
27 |
14 7
|
mndidcl |
|- ( R e. Mnd -> .0. e. ( Base ` R ) ) |
28 |
6 27
|
syl |
|- ( ph -> .0. e. ( Base ` R ) ) |
29 |
28 2
|
eleqtrrd |
|- ( ph -> .0. e. V ) |
30 |
2
|
eleq2d |
|- ( ph -> ( x e. V <-> x e. ( Base ` R ) ) ) |
31 |
30
|
biimpa |
|- ( ( ph /\ x e. V ) -> x e. ( Base ` R ) ) |
32 |
14 3 7
|
mndlid |
|- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( .0. .+ x ) = x ) |
33 |
6 31 32
|
syl2an2r |
|- ( ( ph /\ x e. V ) -> ( .0. .+ x ) = x ) |
34 |
33
|
fveq2d |
|- ( ( ph /\ x e. V ) -> ( F ` ( .0. .+ x ) ) = ( F ` x ) ) |
35 |
14 3 7
|
mndrid |
|- ( ( R e. Mnd /\ x e. ( Base ` R ) ) -> ( x .+ .0. ) = x ) |
36 |
6 31 35
|
syl2an2r |
|- ( ( ph /\ x e. V ) -> ( x .+ .0. ) = x ) |
37 |
36
|
fveq2d |
|- ( ( ph /\ x e. V ) -> ( F ` ( x .+ .0. ) ) = ( F ` x ) ) |
38 |
1 2 3 4 5 6 17 26 29 34 37
|
imasmnd2 |
|- ( ph -> ( U e. Mnd /\ ( F ` .0. ) = ( 0g ` U ) ) ) |