Step |
Hyp |
Ref |
Expression |
1 |
|
gsumress.b |
|- B = ( Base ` G ) |
2 |
|
gsumress.o |
|- .+ = ( +g ` G ) |
3 |
|
gsumress.h |
|- H = ( G |`s S ) |
4 |
|
gsumress.g |
|- ( ph -> G e. V ) |
5 |
|
gsumress.a |
|- ( ph -> A e. X ) |
6 |
|
gsumress.s |
|- ( ph -> S C_ B ) |
7 |
|
gsumress.f |
|- ( ph -> F : A --> S ) |
8 |
|
gsumress.z |
|- ( ph -> .0. e. S ) |
9 |
|
gsumress.c |
|- ( ( ph /\ x e. B ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) |
10 |
|
oveq1 |
|- ( y = .0. -> ( y .+ x ) = ( .0. .+ x ) ) |
11 |
10
|
eqeq1d |
|- ( y = .0. -> ( ( y .+ x ) = x <-> ( .0. .+ x ) = x ) ) |
12 |
11
|
ovanraleqv |
|- ( y = .0. -> ( A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) ) |
13 |
6 8
|
sseldd |
|- ( ph -> .0. e. B ) |
14 |
9
|
ralrimiva |
|- ( ph -> A. x e. B ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) |
15 |
12 13 14
|
elrabd |
|- ( ph -> .0. e. { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) |
16 |
15
|
snssd |
|- ( ph -> { .0. } C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) |
17 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
18 |
|
eqid |
|- { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } = { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } |
19 |
1 17 2 18
|
mgmidsssn0 |
|- ( G e. V -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } ) |
20 |
4 19
|
syl |
|- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { ( 0g ` G ) } ) |
21 |
20 15
|
sseldd |
|- ( ph -> .0. e. { ( 0g ` G ) } ) |
22 |
|
elsni |
|- ( .0. e. { ( 0g ` G ) } -> .0. = ( 0g ` G ) ) |
23 |
21 22
|
syl |
|- ( ph -> .0. = ( 0g ` G ) ) |
24 |
23
|
sneqd |
|- ( ph -> { .0. } = { ( 0g ` G ) } ) |
25 |
20 24
|
sseqtrrd |
|- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } C_ { .0. } ) |
26 |
16 25
|
eqssd |
|- ( ph -> { .0. } = { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) |
27 |
11
|
ovanraleqv |
|- ( y = .0. -> ( A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) ) |
28 |
6
|
sselda |
|- ( ( ph /\ x e. S ) -> x e. B ) |
29 |
28 9
|
syldan |
|- ( ( ph /\ x e. S ) -> ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) |
30 |
29
|
ralrimiva |
|- ( ph -> A. x e. S ( ( .0. .+ x ) = x /\ ( x .+ .0. ) = x ) ) |
31 |
27 8 30
|
elrabd |
|- ( ph -> .0. e. { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) |
32 |
3 1
|
ressbas2 |
|- ( S C_ B -> S = ( Base ` H ) ) |
33 |
6 32
|
syl |
|- ( ph -> S = ( Base ` H ) ) |
34 |
|
fvex |
|- ( Base ` H ) e. _V |
35 |
33 34
|
eqeltrdi |
|- ( ph -> S e. _V ) |
36 |
3 2
|
ressplusg |
|- ( S e. _V -> .+ = ( +g ` H ) ) |
37 |
35 36
|
syl |
|- ( ph -> .+ = ( +g ` H ) ) |
38 |
37
|
oveqd |
|- ( ph -> ( y .+ x ) = ( y ( +g ` H ) x ) ) |
39 |
38
|
eqeq1d |
|- ( ph -> ( ( y .+ x ) = x <-> ( y ( +g ` H ) x ) = x ) ) |
40 |
37
|
oveqd |
|- ( ph -> ( x .+ y ) = ( x ( +g ` H ) y ) ) |
41 |
40
|
eqeq1d |
|- ( ph -> ( ( x .+ y ) = x <-> ( x ( +g ` H ) y ) = x ) ) |
42 |
39 41
|
anbi12d |
|- ( ph -> ( ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) ) |
43 |
33 42
|
raleqbidv |
|- ( ph -> ( A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) <-> A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) ) ) |
44 |
33 43
|
rabeqbidv |
|- ( ph -> { y e. S | A. x e. S ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) |
45 |
31 44
|
eleqtrd |
|- ( ph -> .0. e. { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) |
46 |
45
|
snssd |
|- ( ph -> { .0. } C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) |
47 |
3
|
ovexi |
|- H e. _V |
48 |
47
|
a1i |
|- ( ph -> H e. _V ) |
49 |
|
eqid |
|- ( Base ` H ) = ( Base ` H ) |
50 |
|
eqid |
|- ( 0g ` H ) = ( 0g ` H ) |
51 |
|
eqid |
|- ( +g ` H ) = ( +g ` H ) |
52 |
|
eqid |
|- { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } |
53 |
49 50 51 52
|
mgmidsssn0 |
|- ( H e. _V -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { ( 0g ` H ) } ) |
54 |
48 53
|
syl |
|- ( ph -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { ( 0g ` H ) } ) |
55 |
54 45
|
sseldd |
|- ( ph -> .0. e. { ( 0g ` H ) } ) |
56 |
|
elsni |
|- ( .0. e. { ( 0g ` H ) } -> .0. = ( 0g ` H ) ) |
57 |
55 56
|
syl |
|- ( ph -> .0. = ( 0g ` H ) ) |
58 |
57
|
sneqd |
|- ( ph -> { .0. } = { ( 0g ` H ) } ) |
59 |
54 58
|
sseqtrrd |
|- ( ph -> { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } C_ { .0. } ) |
60 |
46 59
|
eqssd |
|- ( ph -> { .0. } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) |
61 |
26 60
|
eqtr3d |
|- ( ph -> { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } = { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) |
62 |
61
|
sseq2d |
|- ( ph -> ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } <-> ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) ) |
63 |
23 57
|
eqtr3d |
|- ( ph -> ( 0g ` G ) = ( 0g ` H ) ) |
64 |
37
|
seqeq2d |
|- ( ph -> seq m ( .+ , F ) = seq m ( ( +g ` H ) , F ) ) |
65 |
64
|
fveq1d |
|- ( ph -> ( seq m ( .+ , F ) ` n ) = ( seq m ( ( +g ` H ) , F ) ` n ) ) |
66 |
65
|
eqeq2d |
|- ( ph -> ( z = ( seq m ( .+ , F ) ` n ) <-> z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) |
67 |
66
|
anbi2d |
|- ( ph -> ( ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) |
68 |
67
|
rexbidv |
|- ( ph -> ( E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) |
69 |
68
|
exbidv |
|- ( ph -> ( E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) <-> E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) |
70 |
69
|
iotabidv |
|- ( ph -> ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) = ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) ) |
71 |
37
|
seqeq2d |
|- ( ph -> seq 1 ( .+ , ( F o. f ) ) = seq 1 ( ( +g ` H ) , ( F o. f ) ) ) |
72 |
71
|
fveq1d |
|- ( ph -> ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) |
73 |
72
|
eqeq2d |
|- ( ph -> ( z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) <-> z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) |
74 |
73
|
anbi2d |
|- ( ph -> ( ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) <-> ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) |
75 |
74
|
exbidv |
|- ( ph -> ( E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) <-> E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) |
76 |
75
|
iotabidv |
|- ( ph -> ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) = ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) |
77 |
70 76
|
ifeq12d |
|- ( ph -> if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) = if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) |
78 |
62 63 77
|
ifbieq12d |
|- ( ph -> if ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } , ( 0g ` G ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) = if ( ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } , ( 0g ` H ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) ) |
79 |
26
|
difeq2d |
|- ( ph -> ( _V \ { .0. } ) = ( _V \ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) ) |
80 |
79
|
imaeq2d |
|- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( `' F " ( _V \ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } ) ) ) |
81 |
7 6
|
fssd |
|- ( ph -> F : A --> B ) |
82 |
1 17 2 18 80 4 5 81
|
gsumval |
|- ( ph -> ( G gsum F ) = if ( ran F C_ { y e. B | A. x e. B ( ( y .+ x ) = x /\ ( x .+ y ) = x ) } , ( 0g ` G ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( .+ , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( .+ , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) ) |
83 |
60
|
difeq2d |
|- ( ph -> ( _V \ { .0. } ) = ( _V \ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) ) |
84 |
83
|
imaeq2d |
|- ( ph -> ( `' F " ( _V \ { .0. } ) ) = ( `' F " ( _V \ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } ) ) ) |
85 |
33
|
feq3d |
|- ( ph -> ( F : A --> S <-> F : A --> ( Base ` H ) ) ) |
86 |
7 85
|
mpbid |
|- ( ph -> F : A --> ( Base ` H ) ) |
87 |
49 50 51 52 84 48 5 86
|
gsumval |
|- ( ph -> ( H gsum F ) = if ( ran F C_ { y e. ( Base ` H ) | A. x e. ( Base ` H ) ( ( y ( +g ` H ) x ) = x /\ ( x ( +g ` H ) y ) = x ) } , ( 0g ` H ) , if ( A e. ran ... , ( iota z E. m E. n e. ( ZZ>= ` m ) ( A = ( m ... n ) /\ z = ( seq m ( ( +g ` H ) , F ) ` n ) ) ) , ( iota z E. f ( f : ( 1 ... ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) -1-1-onto-> ( `' F " ( _V \ { .0. } ) ) /\ z = ( seq 1 ( ( +g ` H ) , ( F o. f ) ) ` ( # ` ( `' F " ( _V \ { .0. } ) ) ) ) ) ) ) ) ) |
88 |
78 82 87
|
3eqtr4d |
|- ( ph -> ( G gsum F ) = ( H gsum F ) ) |