| Step |
Hyp |
Ref |
Expression |
| 1 |
|
zrhcntr.1 |
|- M = ( mulGrp ` R ) |
| 2 |
|
zrhcntr.2 |
|- C = ( Cntr ` M ) |
| 3 |
|
zrhcntr.3 |
|- L = ( ZRHom ` R ) |
| 4 |
|
zrhcntr.4 |
|- ( ph -> R e. Ring ) |
| 5 |
|
zrhcntr.5 |
|- ( ph -> N e. ZZ ) |
| 6 |
|
fveq2 |
|- ( m = N -> ( L ` m ) = ( L ` N ) ) |
| 7 |
6
|
eleq1d |
|- ( m = N -> ( ( L ` m ) e. C <-> ( L ` N ) e. C ) ) |
| 8 |
|
fveq2 |
|- ( i = 0 -> ( L ` i ) = ( L ` 0 ) ) |
| 9 |
8
|
eleq1d |
|- ( i = 0 -> ( ( L ` i ) e. C <-> ( L ` 0 ) e. C ) ) |
| 10 |
|
fveq2 |
|- ( i = n -> ( L ` i ) = ( L ` n ) ) |
| 11 |
10
|
eleq1d |
|- ( i = n -> ( ( L ` i ) e. C <-> ( L ` n ) e. C ) ) |
| 12 |
|
fveq2 |
|- ( i = ( n + 1 ) -> ( L ` i ) = ( L ` ( n + 1 ) ) ) |
| 13 |
12
|
eleq1d |
|- ( i = ( n + 1 ) -> ( ( L ` i ) e. C <-> ( L ` ( n + 1 ) ) e. C ) ) |
| 14 |
|
fveq2 |
|- ( i = m -> ( L ` i ) = ( L ` m ) ) |
| 15 |
14
|
eleq1d |
|- ( i = m -> ( ( L ` i ) e. C <-> ( L ` m ) e. C ) ) |
| 16 |
|
eqid |
|- ( 0g ` R ) = ( 0g ` R ) |
| 17 |
3 16
|
zrh0 |
|- ( R e. Ring -> ( L ` 0 ) = ( 0g ` R ) ) |
| 18 |
4 17
|
syl |
|- ( ph -> ( L ` 0 ) = ( 0g ` R ) ) |
| 19 |
|
eqid |
|- ( Base ` R ) = ( Base ` R ) |
| 20 |
19 16
|
ring0cl |
|- ( R e. Ring -> ( 0g ` R ) e. ( Base ` R ) ) |
| 21 |
4 20
|
syl |
|- ( ph -> ( 0g ` R ) e. ( Base ` R ) ) |
| 22 |
18 21
|
eqeltrd |
|- ( ph -> ( L ` 0 ) e. ( Base ` R ) ) |
| 23 |
|
eqid |
|- ( .r ` R ) = ( .r ` R ) |
| 24 |
4
|
adantr |
|- ( ( ph /\ x e. ( Base ` R ) ) -> R e. Ring ) |
| 25 |
|
simpr |
|- ( ( ph /\ x e. ( Base ` R ) ) -> x e. ( Base ` R ) ) |
| 26 |
19 23 16 24 25
|
ringlzd |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( ( 0g ` R ) ( .r ` R ) x ) = ( 0g ` R ) ) |
| 27 |
19 23 16 24 25
|
ringrzd |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( 0g ` R ) ) = ( 0g ` R ) ) |
| 28 |
26 27
|
eqtr4d |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( ( 0g ` R ) ( .r ` R ) x ) = ( x ( .r ` R ) ( 0g ` R ) ) ) |
| 29 |
18
|
oveq1d |
|- ( ph -> ( ( L ` 0 ) ( .r ` R ) x ) = ( ( 0g ` R ) ( .r ` R ) x ) ) |
| 30 |
29
|
adantr |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( ( L ` 0 ) ( .r ` R ) x ) = ( ( 0g ` R ) ( .r ` R ) x ) ) |
| 31 |
18
|
oveq2d |
|- ( ph -> ( x ( .r ` R ) ( L ` 0 ) ) = ( x ( .r ` R ) ( 0g ` R ) ) ) |
| 32 |
31
|
adantr |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( L ` 0 ) ) = ( x ( .r ` R ) ( 0g ` R ) ) ) |
| 33 |
28 30 32
|
3eqtr4d |
|- ( ( ph /\ x e. ( Base ` R ) ) -> ( ( L ` 0 ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` 0 ) ) ) |
| 34 |
33
|
ralrimiva |
|- ( ph -> A. x e. ( Base ` R ) ( ( L ` 0 ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` 0 ) ) ) |
| 35 |
1 19
|
mgpbas |
|- ( Base ` R ) = ( Base ` M ) |
| 36 |
1 23
|
mgpplusg |
|- ( .r ` R ) = ( +g ` M ) |
| 37 |
35 36 2
|
elcntr |
|- ( ( L ` 0 ) e. C <-> ( ( L ` 0 ) e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( L ` 0 ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` 0 ) ) ) ) |
| 38 |
22 34 37
|
sylanbrc |
|- ( ph -> ( L ` 0 ) e. C ) |
| 39 |
3
|
zrhrhm |
|- ( R e. Ring -> L e. ( ZZring RingHom R ) ) |
| 40 |
|
rhmghm |
|- ( L e. ( ZZring RingHom R ) -> L e. ( ZZring GrpHom R ) ) |
| 41 |
4 39 40
|
3syl |
|- ( ph -> L e. ( ZZring GrpHom R ) ) |
| 42 |
41
|
ad2antrr |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> L e. ( ZZring GrpHom R ) ) |
| 43 |
|
simplr |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> n e. NN0 ) |
| 44 |
43
|
nn0zd |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> n e. ZZ ) |
| 45 |
|
1zzd |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> 1 e. ZZ ) |
| 46 |
|
zringbas |
|- ZZ = ( Base ` ZZring ) |
| 47 |
|
zringplusg |
|- + = ( +g ` ZZring ) |
| 48 |
|
eqid |
|- ( +g ` R ) = ( +g ` R ) |
| 49 |
46 47 48
|
ghmlin |
|- ( ( L e. ( ZZring GrpHom R ) /\ n e. ZZ /\ 1 e. ZZ ) -> ( L ` ( n + 1 ) ) = ( ( L ` n ) ( +g ` R ) ( L ` 1 ) ) ) |
| 50 |
42 44 45 49
|
syl3anc |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( L ` ( n + 1 ) ) = ( ( L ` n ) ( +g ` R ) ( L ` 1 ) ) ) |
| 51 |
|
eqid |
|- ( 1r ` R ) = ( 1r ` R ) |
| 52 |
3 51
|
zrh1 |
|- ( R e. Ring -> ( L ` 1 ) = ( 1r ` R ) ) |
| 53 |
4 52
|
syl |
|- ( ph -> ( L ` 1 ) = ( 1r ` R ) ) |
| 54 |
53
|
ad2antrr |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( L ` 1 ) = ( 1r ` R ) ) |
| 55 |
54
|
oveq2d |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( ( L ` n ) ( +g ` R ) ( L ` 1 ) ) = ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) |
| 56 |
50 55
|
eqtrd |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( L ` ( n + 1 ) ) = ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) |
| 57 |
4
|
ringgrpd |
|- ( ph -> R e. Grp ) |
| 58 |
57
|
ad2antrr |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> R e. Grp ) |
| 59 |
35
|
cntrss |
|- ( Cntr ` M ) C_ ( Base ` R ) |
| 60 |
2 59
|
eqsstri |
|- C C_ ( Base ` R ) |
| 61 |
60
|
a1i |
|- ( ( ph /\ n e. NN0 ) -> C C_ ( Base ` R ) ) |
| 62 |
61
|
sselda |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( L ` n ) e. ( Base ` R ) ) |
| 63 |
19 51
|
ringidcl |
|- ( R e. Ring -> ( 1r ` R ) e. ( Base ` R ) ) |
| 64 |
4 63
|
syl |
|- ( ph -> ( 1r ` R ) e. ( Base ` R ) ) |
| 65 |
64
|
ad2antrr |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( 1r ` R ) e. ( Base ` R ) ) |
| 66 |
19 48 58 62 65
|
grpcld |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) e. ( Base ` R ) ) |
| 67 |
35 36 2
|
cntri |
|- ( ( ( L ` n ) e. C /\ x e. ( Base ` R ) ) -> ( ( L ` n ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` n ) ) ) |
| 68 |
67
|
adantll |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( L ` n ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` n ) ) ) |
| 69 |
4
|
ad3antrrr |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> R e. Ring ) |
| 70 |
|
simpr |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> x e. ( Base ` R ) ) |
| 71 |
19 23 51 69 70
|
ringlidmd |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = x ) |
| 72 |
19 23 51 69 70
|
ringridmd |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( 1r ` R ) ) = x ) |
| 73 |
71 72
|
eqtr4d |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( 1r ` R ) ( .r ` R ) x ) = ( x ( .r ` R ) ( 1r ` R ) ) ) |
| 74 |
68 73
|
oveq12d |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( ( L ` n ) ( .r ` R ) x ) ( +g ` R ) ( ( 1r ` R ) ( .r ` R ) x ) ) = ( ( x ( .r ` R ) ( L ` n ) ) ( +g ` R ) ( x ( .r ` R ) ( 1r ` R ) ) ) ) |
| 75 |
62
|
adantr |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( L ` n ) e. ( Base ` R ) ) |
| 76 |
69 63
|
syl |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( 1r ` R ) e. ( Base ` R ) ) |
| 77 |
19 48 23 69 75 76 70
|
ringdird |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ( .r ` R ) x ) = ( ( ( L ` n ) ( .r ` R ) x ) ( +g ` R ) ( ( 1r ` R ) ( .r ` R ) x ) ) ) |
| 78 |
19 48 23 69 70 75 76
|
ringdid |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) = ( ( x ( .r ` R ) ( L ` n ) ) ( +g ` R ) ( x ( .r ` R ) ( 1r ` R ) ) ) ) |
| 79 |
74 77 78
|
3eqtr4d |
|- ( ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) /\ x e. ( Base ` R ) ) -> ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ( .r ` R ) x ) = ( x ( .r ` R ) ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) ) |
| 80 |
79
|
ralrimiva |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> A. x e. ( Base ` R ) ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ( .r ` R ) x ) = ( x ( .r ` R ) ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) ) |
| 81 |
35 36 2
|
elcntr |
|- ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) e. C <-> ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ( .r ` R ) x ) = ( x ( .r ` R ) ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) ) ) ) |
| 82 |
66 80 81
|
sylanbrc |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( ( L ` n ) ( +g ` R ) ( 1r ` R ) ) e. C ) |
| 83 |
56 82
|
eqeltrd |
|- ( ( ( ph /\ n e. NN0 ) /\ ( L ` n ) e. C ) -> ( L ` ( n + 1 ) ) e. C ) |
| 84 |
9 11 13 15 38 83
|
nn0indd |
|- ( ( ph /\ m e. NN0 ) -> ( L ` m ) e. C ) |
| 85 |
84
|
ralrimiva |
|- ( ph -> A. m e. NN0 ( L ` m ) e. C ) |
| 86 |
85
|
adantr |
|- ( ( ph /\ N e. NN0 ) -> A. m e. NN0 ( L ` m ) e. C ) |
| 87 |
|
simpr |
|- ( ( ph /\ N e. NN0 ) -> N e. NN0 ) |
| 88 |
7 86 87
|
rspcdva |
|- ( ( ph /\ N e. NN0 ) -> ( L ` N ) e. C ) |
| 89 |
46 19
|
rhmf |
|- ( L e. ( ZZring RingHom R ) -> L : ZZ --> ( Base ` R ) ) |
| 90 |
4 39 89
|
3syl |
|- ( ph -> L : ZZ --> ( Base ` R ) ) |
| 91 |
90
|
adantr |
|- ( ( ph /\ -u N e. NN0 ) -> L : ZZ --> ( Base ` R ) ) |
| 92 |
5
|
adantr |
|- ( ( ph /\ -u N e. NN0 ) -> N e. ZZ ) |
| 93 |
91 92
|
ffvelcdmd |
|- ( ( ph /\ -u N e. NN0 ) -> ( L ` N ) e. ( Base ` R ) ) |
| 94 |
|
eqid |
|- ( invg ` R ) = ( invg ` R ) |
| 95 |
4
|
ad2antrr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> R e. Ring ) |
| 96 |
|
fveq2 |
|- ( m = -u N -> ( L ` m ) = ( L ` -u N ) ) |
| 97 |
96
|
eleq1d |
|- ( m = -u N -> ( ( L ` m ) e. C <-> ( L ` -u N ) e. C ) ) |
| 98 |
85
|
adantr |
|- ( ( ph /\ -u N e. NN0 ) -> A. m e. NN0 ( L ` m ) e. C ) |
| 99 |
|
simpr |
|- ( ( ph /\ -u N e. NN0 ) -> -u N e. NN0 ) |
| 100 |
97 98 99
|
rspcdva |
|- ( ( ph /\ -u N e. NN0 ) -> ( L ` -u N ) e. C ) |
| 101 |
35 36 2
|
elcntr |
|- ( ( L ` -u N ) e. C <-> ( ( L ` -u N ) e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( L ` -u N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` -u N ) ) ) ) |
| 102 |
100 101
|
sylib |
|- ( ( ph /\ -u N e. NN0 ) -> ( ( L ` -u N ) e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( L ` -u N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` -u N ) ) ) ) |
| 103 |
102
|
simpld |
|- ( ( ph /\ -u N e. NN0 ) -> ( L ` -u N ) e. ( Base ` R ) ) |
| 104 |
103
|
adantr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( L ` -u N ) e. ( Base ` R ) ) |
| 105 |
|
simpr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> x e. ( Base ` R ) ) |
| 106 |
19 23 94 95 104 105
|
ringmneg1 |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( ( ( invg ` R ) ` ( L ` -u N ) ) ( .r ` R ) x ) = ( ( invg ` R ) ` ( ( L ` -u N ) ( .r ` R ) x ) ) ) |
| 107 |
5
|
zcnd |
|- ( ph -> N e. CC ) |
| 108 |
107
|
ad2antrr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> N e. CC ) |
| 109 |
108
|
negnegd |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> -u -u N = N ) |
| 110 |
5
|
znegcld |
|- ( ph -> -u N e. ZZ ) |
| 111 |
|
zringinvg |
|- ( -u N e. ZZ -> -u -u N = ( ( invg ` ZZring ) ` -u N ) ) |
| 112 |
110 111
|
syl |
|- ( ph -> -u -u N = ( ( invg ` ZZring ) ` -u N ) ) |
| 113 |
112
|
ad2antrr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> -u -u N = ( ( invg ` ZZring ) ` -u N ) ) |
| 114 |
109 113
|
eqtr3d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> N = ( ( invg ` ZZring ) ` -u N ) ) |
| 115 |
114
|
fveq2d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( L ` N ) = ( L ` ( ( invg ` ZZring ) ` -u N ) ) ) |
| 116 |
95 39 40
|
3syl |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> L e. ( ZZring GrpHom R ) ) |
| 117 |
110
|
ad2antrr |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> -u N e. ZZ ) |
| 118 |
|
eqid |
|- ( invg ` ZZring ) = ( invg ` ZZring ) |
| 119 |
46 118 94
|
ghminv |
|- ( ( L e. ( ZZring GrpHom R ) /\ -u N e. ZZ ) -> ( L ` ( ( invg ` ZZring ) ` -u N ) ) = ( ( invg ` R ) ` ( L ` -u N ) ) ) |
| 120 |
116 117 119
|
syl2anc |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( L ` ( ( invg ` ZZring ) ` -u N ) ) = ( ( invg ` R ) ` ( L ` -u N ) ) ) |
| 121 |
115 120
|
eqtrd |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( L ` N ) = ( ( invg ` R ) ` ( L ` -u N ) ) ) |
| 122 |
121
|
oveq1d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( ( L ` N ) ( .r ` R ) x ) = ( ( ( invg ` R ) ` ( L ` -u N ) ) ( .r ` R ) x ) ) |
| 123 |
19 23 94 95 105 104
|
ringmneg2 |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( ( invg ` R ) ` ( L ` -u N ) ) ) = ( ( invg ` R ) ` ( x ( .r ` R ) ( L ` -u N ) ) ) ) |
| 124 |
121
|
oveq2d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( L ` N ) ) = ( x ( .r ` R ) ( ( invg ` R ) ` ( L ` -u N ) ) ) ) |
| 125 |
102
|
simprd |
|- ( ( ph /\ -u N e. NN0 ) -> A. x e. ( Base ` R ) ( ( L ` -u N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` -u N ) ) ) |
| 126 |
125
|
r19.21bi |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( ( L ` -u N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` -u N ) ) ) |
| 127 |
126
|
fveq2d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( ( invg ` R ) ` ( ( L ` -u N ) ( .r ` R ) x ) ) = ( ( invg ` R ) ` ( x ( .r ` R ) ( L ` -u N ) ) ) ) |
| 128 |
123 124 127
|
3eqtr4d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( x ( .r ` R ) ( L ` N ) ) = ( ( invg ` R ) ` ( ( L ` -u N ) ( .r ` R ) x ) ) ) |
| 129 |
106 122 128
|
3eqtr4d |
|- ( ( ( ph /\ -u N e. NN0 ) /\ x e. ( Base ` R ) ) -> ( ( L ` N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` N ) ) ) |
| 130 |
129
|
ralrimiva |
|- ( ( ph /\ -u N e. NN0 ) -> A. x e. ( Base ` R ) ( ( L ` N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` N ) ) ) |
| 131 |
35 36 2
|
elcntr |
|- ( ( L ` N ) e. C <-> ( ( L ` N ) e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( L ` N ) ( .r ` R ) x ) = ( x ( .r ` R ) ( L ` N ) ) ) ) |
| 132 |
93 130 131
|
sylanbrc |
|- ( ( ph /\ -u N e. NN0 ) -> ( L ` N ) e. C ) |
| 133 |
|
elznn0 |
|- ( N e. ZZ <-> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) ) |
| 134 |
5 133
|
sylib |
|- ( ph -> ( N e. RR /\ ( N e. NN0 \/ -u N e. NN0 ) ) ) |
| 135 |
134
|
simprd |
|- ( ph -> ( N e. NN0 \/ -u N e. NN0 ) ) |
| 136 |
88 132 135
|
mpjaodan |
|- ( ph -> ( L ` N ) e. C ) |