Step |
Hyp |
Ref |
Expression |
1 |
|
mndinvmod.b |
|- B = ( Base ` G ) |
2 |
|
mndinvmod.0 |
|- .0. = ( 0g ` G ) |
3 |
|
mndinvmod.p |
|- .+ = ( +g ` G ) |
4 |
|
mndinvmod.m |
|- ( ph -> G e. Mnd ) |
5 |
|
mndinvmod.a |
|- ( ph -> A e. B ) |
6 |
|
simpl |
|- ( ( w e. B /\ v e. B ) -> w e. B ) |
7 |
1 3 2
|
mndrid |
|- ( ( G e. Mnd /\ w e. B ) -> ( w .+ .0. ) = w ) |
8 |
4 6 7
|
syl2an |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( w .+ .0. ) = w ) |
9 |
8
|
eqcomd |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> w = ( w .+ .0. ) ) |
10 |
9
|
adantr |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> w = ( w .+ .0. ) ) |
11 |
|
oveq2 |
|- ( .0. = ( A .+ v ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) ) |
12 |
11
|
eqcoms |
|- ( ( A .+ v ) = .0. -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) ) |
13 |
12
|
adantl |
|- ( ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) ) |
14 |
13
|
adantl |
|- ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) ) |
15 |
14
|
adantl |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ .0. ) = ( w .+ ( A .+ v ) ) ) |
16 |
4
|
adantr |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> G e. Mnd ) |
17 |
6
|
adantl |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> w e. B ) |
18 |
5
|
adantr |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> A e. B ) |
19 |
|
simpr |
|- ( ( w e. B /\ v e. B ) -> v e. B ) |
20 |
19
|
adantl |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> v e. B ) |
21 |
1 3
|
mndass |
|- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( ( w .+ A ) .+ v ) = ( w .+ ( A .+ v ) ) ) |
22 |
21
|
eqcomd |
|- ( ( G e. Mnd /\ ( w e. B /\ A e. B /\ v e. B ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) ) |
23 |
16 17 18 20 22
|
syl13anc |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) ) |
24 |
23
|
adantr |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ ( A .+ v ) ) = ( ( w .+ A ) .+ v ) ) |
25 |
|
oveq1 |
|- ( ( w .+ A ) = .0. -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) ) |
26 |
25
|
adantr |
|- ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) ) |
27 |
26
|
adantr |
|- ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) ) |
28 |
27
|
adantl |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( ( w .+ A ) .+ v ) = ( .0. .+ v ) ) |
29 |
1 3 2
|
mndlid |
|- ( ( G e. Mnd /\ v e. B ) -> ( .0. .+ v ) = v ) |
30 |
4 19 29
|
syl2an |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( .0. .+ v ) = v ) |
31 |
30
|
adantr |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( .0. .+ v ) = v ) |
32 |
24 28 31
|
3eqtrd |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> ( w .+ ( A .+ v ) ) = v ) |
33 |
10 15 32
|
3eqtrd |
|- ( ( ( ph /\ ( w e. B /\ v e. B ) ) /\ ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) -> w = v ) |
34 |
33
|
ex |
|- ( ( ph /\ ( w e. B /\ v e. B ) ) -> ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) ) |
35 |
34
|
ralrimivva |
|- ( ph -> A. w e. B A. v e. B ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) ) |
36 |
|
oveq1 |
|- ( w = v -> ( w .+ A ) = ( v .+ A ) ) |
37 |
36
|
eqeq1d |
|- ( w = v -> ( ( w .+ A ) = .0. <-> ( v .+ A ) = .0. ) ) |
38 |
|
oveq2 |
|- ( w = v -> ( A .+ w ) = ( A .+ v ) ) |
39 |
38
|
eqeq1d |
|- ( w = v -> ( ( A .+ w ) = .0. <-> ( A .+ v ) = .0. ) ) |
40 |
37 39
|
anbi12d |
|- ( w = v -> ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) <-> ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) ) |
41 |
40
|
rmo4 |
|- ( E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) <-> A. w e. B A. v e. B ( ( ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) /\ ( ( v .+ A ) = .0. /\ ( A .+ v ) = .0. ) ) -> w = v ) ) |
42 |
35 41
|
sylibr |
|- ( ph -> E* w e. B ( ( w .+ A ) = .0. /\ ( A .+ w ) = .0. ) ) |