Step |
Hyp |
Ref |
Expression |
1 |
|
mulgval.b |
|- B = ( Base ` G ) |
2 |
|
mulgval.p |
|- .+ = ( +g ` G ) |
3 |
|
mulgval.o |
|- .0. = ( 0g ` G ) |
4 |
|
mulgval.i |
|- I = ( invg ` G ) |
5 |
|
mulgval.t |
|- .x. = ( .g ` G ) |
6 |
|
eqidd |
|- ( w = G -> ZZ = ZZ ) |
7 |
|
fveq2 |
|- ( w = G -> ( Base ` w ) = ( Base ` G ) ) |
8 |
7 1
|
eqtr4di |
|- ( w = G -> ( Base ` w ) = B ) |
9 |
|
fveq2 |
|- ( w = G -> ( 0g ` w ) = ( 0g ` G ) ) |
10 |
9 3
|
eqtr4di |
|- ( w = G -> ( 0g ` w ) = .0. ) |
11 |
|
fvex |
|- ( +g ` w ) e. _V |
12 |
|
1z |
|- 1 e. ZZ |
13 |
11 12
|
seqexw |
|- seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V |
14 |
13
|
a1i |
|- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) e. _V ) |
15 |
|
id |
|- ( s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) -> s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) |
16 |
|
fveq2 |
|- ( w = G -> ( +g ` w ) = ( +g ` G ) ) |
17 |
16 2
|
eqtr4di |
|- ( w = G -> ( +g ` w ) = .+ ) |
18 |
17
|
seqeq2d |
|- ( w = G -> seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) = seq 1 ( .+ , ( NN X. { x } ) ) ) |
19 |
15 18
|
sylan9eqr |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> s = seq 1 ( .+ , ( NN X. { x } ) ) ) |
20 |
19
|
fveq1d |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) ) |
21 |
|
simpl |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> w = G ) |
22 |
21
|
fveq2d |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = ( invg ` G ) ) |
23 |
22 4
|
eqtr4di |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( invg ` w ) = I ) |
24 |
19
|
fveq1d |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( s ` -u n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) |
25 |
23 24
|
fveq12d |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> ( ( invg ` w ) ` ( s ` -u n ) ) = ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) |
26 |
20 25
|
ifeq12d |
|- ( ( w = G /\ s = seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) ) -> if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) = if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) |
27 |
14 26
|
csbied |
|- ( w = G -> [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) = if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) |
28 |
10 27
|
ifeq12d |
|- ( w = G -> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) = if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) |
29 |
6 8 28
|
mpoeq123dv |
|- ( w = G -> ( n e. ZZ , x e. ( Base ` w ) |-> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) ) = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) ) |
30 |
|
df-mulg |
|- .g = ( w e. _V |-> ( n e. ZZ , x e. ( Base ` w ) |-> if ( n = 0 , ( 0g ` w ) , [_ seq 1 ( ( +g ` w ) , ( NN X. { x } ) ) / s ]_ if ( 0 < n , ( s ` n ) , ( ( invg ` w ) ` ( s ` -u n ) ) ) ) ) ) |
31 |
|
zex |
|- ZZ e. _V |
32 |
1
|
fvexi |
|- B e. _V |
33 |
|
snex |
|- { .0. } e. _V |
34 |
2
|
fvexi |
|- .+ e. _V |
35 |
34
|
rnex |
|- ran .+ e. _V |
36 |
35 32
|
unex |
|- ( ran .+ u. B ) e. _V |
37 |
4
|
fvexi |
|- I e. _V |
38 |
37
|
rnex |
|- ran I e. _V |
39 |
|
p0ex |
|- { (/) } e. _V |
40 |
38 39
|
unex |
|- ( ran I u. { (/) } ) e. _V |
41 |
36 40
|
unex |
|- ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) e. _V |
42 |
33 41
|
unex |
|- ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) e. _V |
43 |
|
ssun1 |
|- { .0. } C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
44 |
3
|
fvexi |
|- .0. e. _V |
45 |
44
|
snid |
|- .0. e. { .0. } |
46 |
43 45
|
sselii |
|- .0. e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
47 |
46
|
a1i |
|- ( ( n e. ZZ /\ x e. B ) -> .0. e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
48 |
|
ssun2 |
|- B C_ ( ran .+ u. B ) |
49 |
|
ssun1 |
|- ( ran .+ u. B ) C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) |
50 |
48 49
|
sstri |
|- B C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) |
51 |
|
ssun2 |
|- ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
52 |
50 51
|
sstri |
|- B C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
53 |
|
fveq2 |
|- ( n = 1 -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) ) |
54 |
53
|
adantl |
|- ( ( x e. B /\ n = 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) ) |
55 |
|
seq1 |
|- ( 1 e. ZZ -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) = ( ( NN X. { x } ) ` 1 ) ) |
56 |
12 55
|
ax-mp |
|- ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) = ( ( NN X. { x } ) ` 1 ) |
57 |
|
1nn |
|- 1 e. NN |
58 |
|
vex |
|- x e. _V |
59 |
58
|
fvconst2 |
|- ( 1 e. NN -> ( ( NN X. { x } ) ` 1 ) = x ) |
60 |
57 59
|
ax-mp |
|- ( ( NN X. { x } ) ` 1 ) = x |
61 |
60
|
eleq1i |
|- ( ( ( NN X. { x } ) ` 1 ) e. B <-> x e. B ) |
62 |
61
|
biimpri |
|- ( x e. B -> ( ( NN X. { x } ) ` 1 ) e. B ) |
63 |
56 62
|
eqeltrid |
|- ( x e. B -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) e. B ) |
64 |
63
|
adantr |
|- ( ( x e. B /\ n = 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` 1 ) e. B ) |
65 |
54 64
|
eqeltrd |
|- ( ( x e. B /\ n = 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. B ) |
66 |
52 65
|
sselid |
|- ( ( x e. B /\ n = 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
67 |
66
|
ad4ant24 |
|- ( ( ( ( n e. ZZ /\ x e. B ) /\ n e. ( ZZ>= ` 1 ) ) /\ n = 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
68 |
|
zcn |
|- ( n e. ZZ -> n e. CC ) |
69 |
|
npcan1 |
|- ( n e. CC -> ( ( n - 1 ) + 1 ) = n ) |
70 |
68 69
|
syl |
|- ( n e. ZZ -> ( ( n - 1 ) + 1 ) = n ) |
71 |
70
|
fveq2d |
|- ( n e. ZZ -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( ( n - 1 ) + 1 ) ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) ) |
72 |
71
|
adantr |
|- ( ( n e. ZZ /\ ( n - 1 ) e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( ( n - 1 ) + 1 ) ) = ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) ) |
73 |
|
seqp1 |
|- ( ( n - 1 ) e. ( ZZ>= ` 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( ( n - 1 ) + 1 ) ) = ( ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) .+ ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) ) ) |
74 |
|
ssun1 |
|- ran .+ C_ ( ran .+ u. B ) |
75 |
|
ssun2 |
|- { (/) } C_ ( ran I u. { (/) } ) |
76 |
|
unss12 |
|- ( ( ran .+ C_ ( ran .+ u. B ) /\ { (/) } C_ ( ran I u. { (/) } ) ) -> ( ran .+ u. { (/) } ) C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
77 |
74 75 76
|
mp2an |
|- ( ran .+ u. { (/) } ) C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) |
78 |
77 51
|
sstri |
|- ( ran .+ u. { (/) } ) C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
79 |
|
df-ov |
|- ( ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) .+ ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) ) = ( .+ ` <. ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) , ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) >. ) |
80 |
|
fvrn0 |
|- ( .+ ` <. ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) , ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) >. ) e. ( ran .+ u. { (/) } ) |
81 |
79 80
|
eqeltri |
|- ( ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) .+ ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) ) e. ( ran .+ u. { (/) } ) |
82 |
78 81
|
sselii |
|- ( ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( n - 1 ) ) .+ ( ( NN X. { x } ) ` ( ( n - 1 ) + 1 ) ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
83 |
73 82
|
eqeltrdi |
|- ( ( n - 1 ) e. ( ZZ>= ` 1 ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( ( n - 1 ) + 1 ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
84 |
83
|
adantl |
|- ( ( n e. ZZ /\ ( n - 1 ) e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` ( ( n - 1 ) + 1 ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
85 |
72 84
|
eqeltrrd |
|- ( ( n e. ZZ /\ ( n - 1 ) e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
86 |
85
|
ad4ant14 |
|- ( ( ( ( n e. ZZ /\ x e. B ) /\ n e. ( ZZ>= ` 1 ) ) /\ ( n - 1 ) e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
87 |
|
uzm1 |
|- ( n e. ( ZZ>= ` 1 ) -> ( n = 1 \/ ( n - 1 ) e. ( ZZ>= ` 1 ) ) ) |
88 |
87
|
adantl |
|- ( ( ( n e. ZZ /\ x e. B ) /\ n e. ( ZZ>= ` 1 ) ) -> ( n = 1 \/ ( n - 1 ) e. ( ZZ>= ` 1 ) ) ) |
89 |
67 86 88
|
mpjaodan |
|- ( ( ( n e. ZZ /\ x e. B ) /\ n e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
90 |
|
simpr |
|- ( ( ( n e. ZZ /\ x e. B ) /\ -. n e. ( ZZ>= ` 1 ) ) -> -. n e. ( ZZ>= ` 1 ) ) |
91 |
|
seqfn |
|- ( 1 e. ZZ -> seq 1 ( .+ , ( NN X. { x } ) ) Fn ( ZZ>= ` 1 ) ) |
92 |
12 91
|
ax-mp |
|- seq 1 ( .+ , ( NN X. { x } ) ) Fn ( ZZ>= ` 1 ) |
93 |
92
|
fndmi |
|- dom seq 1 ( .+ , ( NN X. { x } ) ) = ( ZZ>= ` 1 ) |
94 |
93
|
eleq2i |
|- ( n e. dom seq 1 ( .+ , ( NN X. { x } ) ) <-> n e. ( ZZ>= ` 1 ) ) |
95 |
90 94
|
sylnibr |
|- ( ( ( n e. ZZ /\ x e. B ) /\ -. n e. ( ZZ>= ` 1 ) ) -> -. n e. dom seq 1 ( .+ , ( NN X. { x } ) ) ) |
96 |
|
ndmfv |
|- ( -. n e. dom seq 1 ( .+ , ( NN X. { x } ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = (/) ) |
97 |
95 96
|
syl |
|- ( ( ( n e. ZZ /\ x e. B ) /\ -. n e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) = (/) ) |
98 |
|
ssun2 |
|- ( ran I u. { (/) } ) C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) |
99 |
75 98
|
sstri |
|- { (/) } C_ ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) |
100 |
99 51
|
sstri |
|- { (/) } C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
101 |
|
0ex |
|- (/) e. _V |
102 |
101
|
snid |
|- (/) e. { (/) } |
103 |
100 102
|
sselii |
|- (/) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
104 |
103
|
a1i |
|- ( ( ( n e. ZZ /\ x e. B ) /\ -. n e. ( ZZ>= ` 1 ) ) -> (/) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
105 |
97 104
|
eqeltrd |
|- ( ( ( n e. ZZ /\ x e. B ) /\ -. n e. ( ZZ>= ` 1 ) ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
106 |
89 105
|
pm2.61dan |
|- ( ( n e. ZZ /\ x e. B ) -> ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
107 |
98 51
|
sstri |
|- ( ran I u. { (/) } ) C_ ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
108 |
|
fvrn0 |
|- ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) e. ( ran I u. { (/) } ) |
109 |
107 108
|
sselii |
|- ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
110 |
109
|
a1i |
|- ( ( n e. ZZ /\ x e. B ) -> ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
111 |
106 110
|
ifcld |
|- ( ( n e. ZZ /\ x e. B ) -> if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
112 |
47 111
|
ifcld |
|- ( ( n e. ZZ /\ x e. B ) -> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) ) |
113 |
112
|
rgen2 |
|- A. n e. ZZ A. x e. B if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) e. ( { .0. } u. ( ( ran .+ u. B ) u. ( ran I u. { (/) } ) ) ) |
114 |
31 32 42 113
|
mpoexw |
|- ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) e. _V |
115 |
29 30 114
|
fvmpt |
|- ( G e. _V -> ( .g ` G ) = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) ) |
116 |
|
fvprc |
|- ( -. G e. _V -> ( .g ` G ) = (/) ) |
117 |
|
eqid |
|- ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) |
118 |
|
fvex |
|- ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) e. _V |
119 |
|
fvex |
|- ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) e. _V |
120 |
118 119
|
ifex |
|- if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) e. _V |
121 |
44 120
|
ifex |
|- if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) e. _V |
122 |
117 121
|
fnmpoi |
|- ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) Fn ( ZZ X. B ) |
123 |
|
fvprc |
|- ( -. G e. _V -> ( Base ` G ) = (/) ) |
124 |
1 123
|
eqtrid |
|- ( -. G e. _V -> B = (/) ) |
125 |
124
|
xpeq2d |
|- ( -. G e. _V -> ( ZZ X. B ) = ( ZZ X. (/) ) ) |
126 |
|
xp0 |
|- ( ZZ X. (/) ) = (/) |
127 |
125 126
|
eqtrdi |
|- ( -. G e. _V -> ( ZZ X. B ) = (/) ) |
128 |
127
|
fneq2d |
|- ( -. G e. _V -> ( ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) Fn ( ZZ X. B ) <-> ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) Fn (/) ) ) |
129 |
122 128
|
mpbii |
|- ( -. G e. _V -> ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) Fn (/) ) |
130 |
|
fn0 |
|- ( ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) Fn (/) <-> ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) = (/) ) |
131 |
129 130
|
sylib |
|- ( -. G e. _V -> ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) = (/) ) |
132 |
116 131
|
eqtr4d |
|- ( -. G e. _V -> ( .g ` G ) = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) ) |
133 |
115 132
|
pm2.61i |
|- ( .g ` G ) = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) |
134 |
5 133
|
eqtri |
|- .x. = ( n e. ZZ , x e. B |-> if ( n = 0 , .0. , if ( 0 < n , ( seq 1 ( .+ , ( NN X. { x } ) ) ` n ) , ( I ` ( seq 1 ( .+ , ( NN X. { x } ) ) ` -u n ) ) ) ) ) |