Step |
Hyp |
Ref |
Expression |
1 |
|
frgpup.b |
|- B = ( Base ` H ) |
2 |
|
frgpup.n |
|- N = ( invg ` H ) |
3 |
|
frgpup.t |
|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) ) |
4 |
|
frgpup.h |
|- ( ph -> H e. Grp ) |
5 |
|
frgpup.i |
|- ( ph -> I e. V ) |
6 |
|
frgpup.a |
|- ( ph -> F : I --> B ) |
7 |
|
frgpup.w |
|- W = ( _I ` Word ( I X. 2o ) ) |
8 |
|
frgpup.r |
|- .~ = ( ~FG ` I ) |
9 |
|
frgpup.g |
|- G = ( freeGrp ` I ) |
10 |
|
frgpup.x |
|- X = ( Base ` G ) |
11 |
|
frgpup.e |
|- E = ran ( g e. W |-> <. [ g ] .~ , ( H gsum ( T o. g ) ) >. ) |
12 |
|
frgpup.u |
|- U = ( varFGrp ` I ) |
13 |
|
frgpup3.k |
|- ( ph -> K e. ( G GrpHom H ) ) |
14 |
|
frgpup3.e |
|- ( ph -> ( K o. U ) = F ) |
15 |
10 1
|
ghmf |
|- ( K e. ( G GrpHom H ) -> K : X --> B ) |
16 |
|
ffn |
|- ( K : X --> B -> K Fn X ) |
17 |
13 15 16
|
3syl |
|- ( ph -> K Fn X ) |
18 |
1 2 3 4 5 6 7 8 9 10 11
|
frgpup1 |
|- ( ph -> E e. ( G GrpHom H ) ) |
19 |
10 1
|
ghmf |
|- ( E e. ( G GrpHom H ) -> E : X --> B ) |
20 |
|
ffn |
|- ( E : X --> B -> E Fn X ) |
21 |
18 19 20
|
3syl |
|- ( ph -> E Fn X ) |
22 |
|
eqid |
|- ( freeMnd ` ( I X. 2o ) ) = ( freeMnd ` ( I X. 2o ) ) |
23 |
9 22 8
|
frgpval |
|- ( I e. V -> G = ( ( freeMnd ` ( I X. 2o ) ) /s .~ ) ) |
24 |
5 23
|
syl |
|- ( ph -> G = ( ( freeMnd ` ( I X. 2o ) ) /s .~ ) ) |
25 |
|
2on |
|- 2o e. On |
26 |
|
xpexg |
|- ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V ) |
27 |
5 25 26
|
sylancl |
|- ( ph -> ( I X. 2o ) e. _V ) |
28 |
|
wrdexg |
|- ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V ) |
29 |
|
fvi |
|- ( Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) |
30 |
27 28 29
|
3syl |
|- ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) ) |
31 |
7 30
|
syl5eq |
|- ( ph -> W = Word ( I X. 2o ) ) |
32 |
|
eqid |
|- ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |
33 |
22 32
|
frmdbas |
|- ( ( I X. 2o ) e. _V -> ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) ) |
34 |
27 33
|
syl |
|- ( ph -> ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) ) |
35 |
31 34
|
eqtr4d |
|- ( ph -> W = ( Base ` ( freeMnd ` ( I X. 2o ) ) ) ) |
36 |
8
|
fvexi |
|- .~ e. _V |
37 |
36
|
a1i |
|- ( ph -> .~ e. _V ) |
38 |
|
fvexd |
|- ( ph -> ( freeMnd ` ( I X. 2o ) ) e. _V ) |
39 |
24 35 37 38
|
qusbas |
|- ( ph -> ( W /. .~ ) = ( Base ` G ) ) |
40 |
10 39
|
eqtr4id |
|- ( ph -> X = ( W /. .~ ) ) |
41 |
|
eqimss |
|- ( X = ( W /. .~ ) -> X C_ ( W /. .~ ) ) |
42 |
40 41
|
syl |
|- ( ph -> X C_ ( W /. .~ ) ) |
43 |
42
|
sselda |
|- ( ( ph /\ a e. X ) -> a e. ( W /. .~ ) ) |
44 |
|
eqid |
|- ( W /. .~ ) = ( W /. .~ ) |
45 |
|
fveq2 |
|- ( [ t ] .~ = a -> ( K ` [ t ] .~ ) = ( K ` a ) ) |
46 |
|
fveq2 |
|- ( [ t ] .~ = a -> ( E ` [ t ] .~ ) = ( E ` a ) ) |
47 |
45 46
|
eqeq12d |
|- ( [ t ] .~ = a -> ( ( K ` [ t ] .~ ) = ( E ` [ t ] .~ ) <-> ( K ` a ) = ( E ` a ) ) ) |
48 |
|
simpr |
|- ( ( ph /\ t e. W ) -> t e. W ) |
49 |
31
|
adantr |
|- ( ( ph /\ t e. W ) -> W = Word ( I X. 2o ) ) |
50 |
48 49
|
eleqtrd |
|- ( ( ph /\ t e. W ) -> t e. Word ( I X. 2o ) ) |
51 |
|
wrdf |
|- ( t e. Word ( I X. 2o ) -> t : ( 0 ..^ ( # ` t ) ) --> ( I X. 2o ) ) |
52 |
50 51
|
syl |
|- ( ( ph /\ t e. W ) -> t : ( 0 ..^ ( # ` t ) ) --> ( I X. 2o ) ) |
53 |
52
|
ffvelrnda |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> ( t ` n ) e. ( I X. 2o ) ) |
54 |
|
elxp2 |
|- ( ( t ` n ) e. ( I X. 2o ) <-> E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. ) |
55 |
53 54
|
sylib |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. ) |
56 |
|
fveq2 |
|- ( y = i -> ( F ` y ) = ( F ` i ) ) |
57 |
56
|
fveq2d |
|- ( y = i -> ( N ` ( F ` y ) ) = ( N ` ( F ` i ) ) ) |
58 |
56 57
|
ifeq12d |
|- ( y = i -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = if ( z = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) ) |
59 |
|
eqeq1 |
|- ( z = j -> ( z = (/) <-> j = (/) ) ) |
60 |
59
|
ifbid |
|- ( z = j -> if ( z = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) ) |
61 |
|
fvex |
|- ( F ` i ) e. _V |
62 |
|
fvex |
|- ( N ` ( F ` i ) ) e. _V |
63 |
61 62
|
ifex |
|- if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) e. _V |
64 |
58 60 3 63
|
ovmpo |
|- ( ( i e. I /\ j e. 2o ) -> ( i T j ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) ) |
65 |
64
|
adantl |
|- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( i T j ) = if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) ) |
66 |
|
elpri |
|- ( j e. { (/) , 1o } -> ( j = (/) \/ j = 1o ) ) |
67 |
|
df2o3 |
|- 2o = { (/) , 1o } |
68 |
66 67
|
eleq2s |
|- ( j e. 2o -> ( j = (/) \/ j = 1o ) ) |
69 |
14
|
adantr |
|- ( ( ph /\ i e. I ) -> ( K o. U ) = F ) |
70 |
69
|
fveq1d |
|- ( ( ph /\ i e. I ) -> ( ( K o. U ) ` i ) = ( F ` i ) ) |
71 |
8 12 9 10
|
vrgpf |
|- ( I e. V -> U : I --> X ) |
72 |
5 71
|
syl |
|- ( ph -> U : I --> X ) |
73 |
|
fvco3 |
|- ( ( U : I --> X /\ i e. I ) -> ( ( K o. U ) ` i ) = ( K ` ( U ` i ) ) ) |
74 |
72 73
|
sylan |
|- ( ( ph /\ i e. I ) -> ( ( K o. U ) ` i ) = ( K ` ( U ` i ) ) ) |
75 |
70 74
|
eqtr3d |
|- ( ( ph /\ i e. I ) -> ( F ` i ) = ( K ` ( U ` i ) ) ) |
76 |
75
|
adantr |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( F ` i ) = ( K ` ( U ` i ) ) ) |
77 |
|
iftrue |
|- ( j = (/) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( F ` i ) ) |
78 |
77
|
adantl |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( F ` i ) ) |
79 |
|
simpr |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> j = (/) ) |
80 |
79
|
opeq2d |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> <. i , j >. = <. i , (/) >. ) |
81 |
80
|
s1eqd |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> <" <. i , j >. "> = <" <. i , (/) >. "> ) |
82 |
81
|
eceq1d |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> [ <" <. i , j >. "> ] .~ = [ <" <. i , (/) >. "> ] .~ ) |
83 |
8 12
|
vrgpval |
|- ( ( I e. V /\ i e. I ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ ) |
84 |
5 83
|
sylan |
|- ( ( ph /\ i e. I ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ ) |
85 |
84
|
adantr |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( U ` i ) = [ <" <. i , (/) >. "> ] .~ ) |
86 |
82 85
|
eqtr4d |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> [ <" <. i , j >. "> ] .~ = ( U ` i ) ) |
87 |
86
|
fveq2d |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> ( K ` [ <" <. i , j >. "> ] .~ ) = ( K ` ( U ` i ) ) ) |
88 |
76 78 87
|
3eqtr4d |
|- ( ( ( ph /\ i e. I ) /\ j = (/) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
89 |
75
|
fveq2d |
|- ( ( ph /\ i e. I ) -> ( N ` ( F ` i ) ) = ( N ` ( K ` ( U ` i ) ) ) ) |
90 |
72
|
ffvelrnda |
|- ( ( ph /\ i e. I ) -> ( U ` i ) e. X ) |
91 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
92 |
10 91 2
|
ghminv |
|- ( ( K e. ( G GrpHom H ) /\ ( U ` i ) e. X ) -> ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) = ( N ` ( K ` ( U ` i ) ) ) ) |
93 |
13 90 92
|
syl2an2r |
|- ( ( ph /\ i e. I ) -> ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) = ( N ` ( K ` ( U ` i ) ) ) ) |
94 |
89 93
|
eqtr4d |
|- ( ( ph /\ i e. I ) -> ( N ` ( F ` i ) ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) ) |
95 |
94
|
adantr |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( N ` ( F ` i ) ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) ) |
96 |
|
1n0 |
|- 1o =/= (/) |
97 |
|
simpr |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> j = 1o ) |
98 |
97
|
neeq1d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( j =/= (/) <-> 1o =/= (/) ) ) |
99 |
96 98
|
mpbiri |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> j =/= (/) ) |
100 |
|
ifnefalse |
|- ( j =/= (/) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( N ` ( F ` i ) ) ) |
101 |
99 100
|
syl |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( N ` ( F ` i ) ) ) |
102 |
97
|
opeq2d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> <. i , j >. = <. i , 1o >. ) |
103 |
102
|
s1eqd |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> <" <. i , j >. "> = <" <. i , 1o >. "> ) |
104 |
103
|
eceq1d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> [ <" <. i , j >. "> ] .~ = [ <" <. i , 1o >. "> ] .~ ) |
105 |
8 12 9 91
|
vrgpinv |
|- ( ( I e. V /\ i e. I ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ ) |
106 |
5 105
|
sylan |
|- ( ( ph /\ i e. I ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ ) |
107 |
106
|
adantr |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( ( invg ` G ) ` ( U ` i ) ) = [ <" <. i , 1o >. "> ] .~ ) |
108 |
104 107
|
eqtr4d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> [ <" <. i , j >. "> ] .~ = ( ( invg ` G ) ` ( U ` i ) ) ) |
109 |
108
|
fveq2d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> ( K ` [ <" <. i , j >. "> ] .~ ) = ( K ` ( ( invg ` G ) ` ( U ` i ) ) ) ) |
110 |
95 101 109
|
3eqtr4d |
|- ( ( ( ph /\ i e. I ) /\ j = 1o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
111 |
88 110
|
jaodan |
|- ( ( ( ph /\ i e. I ) /\ ( j = (/) \/ j = 1o ) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
112 |
68 111
|
sylan2 |
|- ( ( ( ph /\ i e. I ) /\ j e. 2o ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
113 |
112
|
anasss |
|- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> if ( j = (/) , ( F ` i ) , ( N ` ( F ` i ) ) ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
114 |
65 113
|
eqtrd |
|- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( i T j ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
115 |
|
fveq2 |
|- ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( T ` <. i , j >. ) ) |
116 |
|
df-ov |
|- ( i T j ) = ( T ` <. i , j >. ) |
117 |
115 116
|
eqtr4di |
|- ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( i T j ) ) |
118 |
|
s1eq |
|- ( ( t ` n ) = <. i , j >. -> <" ( t ` n ) "> = <" <. i , j >. "> ) |
119 |
118
|
eceq1d |
|- ( ( t ` n ) = <. i , j >. -> [ <" ( t ` n ) "> ] .~ = [ <" <. i , j >. "> ] .~ ) |
120 |
119
|
fveq2d |
|- ( ( t ` n ) = <. i , j >. -> ( K ` [ <" ( t ` n ) "> ] .~ ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) |
121 |
117 120
|
eqeq12d |
|- ( ( t ` n ) = <. i , j >. -> ( ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) <-> ( i T j ) = ( K ` [ <" <. i , j >. "> ] .~ ) ) ) |
122 |
114 121
|
syl5ibrcom |
|- ( ( ph /\ ( i e. I /\ j e. 2o ) ) -> ( ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) ) |
123 |
122
|
rexlimdvva |
|- ( ph -> ( E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) ) |
124 |
123
|
ad2antrr |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> ( E. i e. I E. j e. 2o ( t ` n ) = <. i , j >. -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) ) |
125 |
55 124
|
mpd |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> ( T ` ( t ` n ) ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) |
126 |
125
|
mpteq2dva |
|- ( ( ph /\ t e. W ) -> ( n e. ( 0 ..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> ( K ` [ <" ( t ` n ) "> ] .~ ) ) ) |
127 |
1 2 3 4 5 6
|
frgpuptf |
|- ( ph -> T : ( I X. 2o ) --> B ) |
128 |
|
fcompt |
|- ( ( T : ( I X. 2o ) --> B /\ t : ( 0 ..^ ( # ` t ) ) --> ( I X. 2o ) ) -> ( T o. t ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) ) |
129 |
127 52 128
|
syl2an2r |
|- ( ( ph /\ t e. W ) -> ( T o. t ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> ( T ` ( t ` n ) ) ) ) |
130 |
53
|
s1cld |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> <" ( t ` n ) "> e. Word ( I X. 2o ) ) |
131 |
31
|
ad2antrr |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> W = Word ( I X. 2o ) ) |
132 |
130 131
|
eleqtrrd |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> <" ( t ` n ) "> e. W ) |
133 |
9 8 7 10
|
frgpeccl |
|- ( <" ( t ` n ) "> e. W -> [ <" ( t ` n ) "> ] .~ e. X ) |
134 |
132 133
|
syl |
|- ( ( ( ph /\ t e. W ) /\ n e. ( 0 ..^ ( # ` t ) ) ) -> [ <" ( t ` n ) "> ] .~ e. X ) |
135 |
52
|
feqmptd |
|- ( ( ph /\ t e. W ) -> t = ( n e. ( 0 ..^ ( # ` t ) ) |-> ( t ` n ) ) ) |
136 |
5
|
adantr |
|- ( ( ph /\ t e. W ) -> I e. V ) |
137 |
136 25 26
|
sylancl |
|- ( ( ph /\ t e. W ) -> ( I X. 2o ) e. _V ) |
138 |
|
eqid |
|- ( varFMnd ` ( I X. 2o ) ) = ( varFMnd ` ( I X. 2o ) ) |
139 |
138
|
vrmdfval |
|- ( ( I X. 2o ) e. _V -> ( varFMnd ` ( I X. 2o ) ) = ( w e. ( I X. 2o ) |-> <" w "> ) ) |
140 |
137 139
|
syl |
|- ( ( ph /\ t e. W ) -> ( varFMnd ` ( I X. 2o ) ) = ( w e. ( I X. 2o ) |-> <" w "> ) ) |
141 |
|
s1eq |
|- ( w = ( t ` n ) -> <" w "> = <" ( t ` n ) "> ) |
142 |
53 135 140 141
|
fmptco |
|- ( ( ph /\ t e. W ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> <" ( t ` n ) "> ) ) |
143 |
|
eqidd |
|- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) = ( w e. W |-> [ w ] .~ ) ) |
144 |
|
eceq1 |
|- ( w = <" ( t ` n ) "> -> [ w ] .~ = [ <" ( t ` n ) "> ] .~ ) |
145 |
132 142 143 144
|
fmptco |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> [ <" ( t ` n ) "> ] .~ ) ) |
146 |
13
|
adantr |
|- ( ( ph /\ t e. W ) -> K e. ( G GrpHom H ) ) |
147 |
146 15
|
syl |
|- ( ( ph /\ t e. W ) -> K : X --> B ) |
148 |
147
|
feqmptd |
|- ( ( ph /\ t e. W ) -> K = ( w e. X |-> ( K ` w ) ) ) |
149 |
|
fveq2 |
|- ( w = [ <" ( t ` n ) "> ] .~ -> ( K ` w ) = ( K ` [ <" ( t ` n ) "> ] .~ ) ) |
150 |
134 145 148 149
|
fmptco |
|- ( ( ph /\ t e. W ) -> ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) = ( n e. ( 0 ..^ ( # ` t ) ) |-> ( K ` [ <" ( t ` n ) "> ] .~ ) ) ) |
151 |
126 129 150
|
3eqtr4d |
|- ( ( ph /\ t e. W ) -> ( T o. t ) = ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) |
152 |
151
|
oveq2d |
|- ( ( ph /\ t e. W ) -> ( H gsum ( T o. t ) ) = ( H gsum ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) ) |
153 |
1 2 3 4 5 6 7 8 9 10 11
|
frgpupval |
|- ( ( ph /\ t e. W ) -> ( E ` [ t ] .~ ) = ( H gsum ( T o. t ) ) ) |
154 |
|
ghmmhm |
|- ( K e. ( G GrpHom H ) -> K e. ( G MndHom H ) ) |
155 |
146 154
|
syl |
|- ( ( ph /\ t e. W ) -> K e. ( G MndHom H ) ) |
156 |
138
|
vrmdf |
|- ( ( I X. 2o ) e. _V -> ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> Word ( I X. 2o ) ) |
157 |
137 156
|
syl |
|- ( ( ph /\ t e. W ) -> ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> Word ( I X. 2o ) ) |
158 |
49
|
feq3d |
|- ( ( ph /\ t e. W ) -> ( ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W <-> ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> Word ( I X. 2o ) ) ) |
159 |
157 158
|
mpbird |
|- ( ( ph /\ t e. W ) -> ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W ) |
160 |
|
wrdco |
|- ( ( t e. Word ( I X. 2o ) /\ ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> W ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word W ) |
161 |
50 159 160
|
syl2anc |
|- ( ( ph /\ t e. W ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word W ) |
162 |
35
|
adantr |
|- ( ( ph /\ t e. W ) -> W = ( Base ` ( freeMnd ` ( I X. 2o ) ) ) ) |
163 |
162
|
mpteq1d |
|- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) = ( w e. ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) ) |
164 |
|
eqid |
|- ( w e. ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) = ( w e. ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) |
165 |
22 32 9 8 164
|
frgpmhm |
|- ( I e. V -> ( w e. ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) e. ( ( freeMnd ` ( I X. 2o ) ) MndHom G ) ) |
166 |
136 165
|
syl |
|- ( ( ph /\ t e. W ) -> ( w e. ( Base ` ( freeMnd ` ( I X. 2o ) ) ) |-> [ w ] .~ ) e. ( ( freeMnd ` ( I X. 2o ) ) MndHom G ) ) |
167 |
163 166
|
eqeltrd |
|- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) e. ( ( freeMnd ` ( I X. 2o ) ) MndHom G ) ) |
168 |
32 10
|
mhmf |
|- ( ( w e. W |-> [ w ] .~ ) e. ( ( freeMnd ` ( I X. 2o ) ) MndHom G ) -> ( w e. W |-> [ w ] .~ ) : ( Base ` ( freeMnd ` ( I X. 2o ) ) ) --> X ) |
169 |
167 168
|
syl |
|- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) : ( Base ` ( freeMnd ` ( I X. 2o ) ) ) --> X ) |
170 |
162
|
feq2d |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) : W --> X <-> ( w e. W |-> [ w ] .~ ) : ( Base ` ( freeMnd ` ( I X. 2o ) ) ) --> X ) ) |
171 |
169 170
|
mpbird |
|- ( ( ph /\ t e. W ) -> ( w e. W |-> [ w ] .~ ) : W --> X ) |
172 |
|
wrdco |
|- ( ( ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word W /\ ( w e. W |-> [ w ] .~ ) : W --> X ) -> ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) e. Word X ) |
173 |
161 171 172
|
syl2anc |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) e. Word X ) |
174 |
10
|
gsumwmhm |
|- ( ( K e. ( G MndHom H ) /\ ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) e. Word X ) -> ( K ` ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) = ( H gsum ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) ) |
175 |
155 173 174
|
syl2anc |
|- ( ( ph /\ t e. W ) -> ( K ` ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) = ( H gsum ( K o. ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) ) |
176 |
152 153 175
|
3eqtr4d |
|- ( ( ph /\ t e. W ) -> ( E ` [ t ] .~ ) = ( K ` ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) ) |
177 |
22 138
|
frmdgsum |
|- ( ( ( I X. 2o ) e. _V /\ t e. Word ( I X. 2o ) ) -> ( ( freeMnd ` ( I X. 2o ) ) gsum ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) = t ) |
178 |
27 50 177
|
syl2an2r |
|- ( ( ph /\ t e. W ) -> ( ( freeMnd ` ( I X. 2o ) ) gsum ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) = t ) |
179 |
178
|
fveq2d |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) ` ( ( freeMnd ` ( I X. 2o ) ) gsum ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) = ( ( w e. W |-> [ w ] .~ ) ` t ) ) |
180 |
|
wrdco |
|- ( ( t e. Word ( I X. 2o ) /\ ( varFMnd ` ( I X. 2o ) ) : ( I X. 2o ) --> Word ( I X. 2o ) ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word Word ( I X. 2o ) ) |
181 |
50 157 180
|
syl2anc |
|- ( ( ph /\ t e. W ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word Word ( I X. 2o ) ) |
182 |
34
|
adantr |
|- ( ( ph /\ t e. W ) -> ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) ) |
183 |
|
wrdeq |
|- ( ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word ( I X. 2o ) -> Word ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word Word ( I X. 2o ) ) |
184 |
182 183
|
syl |
|- ( ( ph /\ t e. W ) -> Word ( Base ` ( freeMnd ` ( I X. 2o ) ) ) = Word Word ( I X. 2o ) ) |
185 |
181 184
|
eleqtrrd |
|- ( ( ph /\ t e. W ) -> ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word ( Base ` ( freeMnd ` ( I X. 2o ) ) ) ) |
186 |
32
|
gsumwmhm |
|- ( ( ( w e. W |-> [ w ] .~ ) e. ( ( freeMnd ` ( I X. 2o ) ) MndHom G ) /\ ( ( varFMnd ` ( I X. 2o ) ) o. t ) e. Word ( Base ` ( freeMnd ` ( I X. 2o ) ) ) ) -> ( ( w e. W |-> [ w ] .~ ) ` ( ( freeMnd ` ( I X. 2o ) ) gsum ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) = ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) |
187 |
167 185 186
|
syl2anc |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) ` ( ( freeMnd ` ( I X. 2o ) ) gsum ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) = ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) |
188 |
7 8
|
efger |
|- .~ Er W |
189 |
188
|
a1i |
|- ( ( ph /\ t e. W ) -> .~ Er W ) |
190 |
7
|
fvexi |
|- W e. _V |
191 |
190
|
a1i |
|- ( ( ph /\ t e. W ) -> W e. _V ) |
192 |
|
eqid |
|- ( w e. W |-> [ w ] .~ ) = ( w e. W |-> [ w ] .~ ) |
193 |
189 191 192
|
divsfval |
|- ( ( ph /\ t e. W ) -> ( ( w e. W |-> [ w ] .~ ) ` t ) = [ t ] .~ ) |
194 |
179 187 193
|
3eqtr3d |
|- ( ( ph /\ t e. W ) -> ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) = [ t ] .~ ) |
195 |
194
|
fveq2d |
|- ( ( ph /\ t e. W ) -> ( K ` ( G gsum ( ( w e. W |-> [ w ] .~ ) o. ( ( varFMnd ` ( I X. 2o ) ) o. t ) ) ) ) = ( K ` [ t ] .~ ) ) |
196 |
176 195
|
eqtr2d |
|- ( ( ph /\ t e. W ) -> ( K ` [ t ] .~ ) = ( E ` [ t ] .~ ) ) |
197 |
44 47 196
|
ectocld |
|- ( ( ph /\ a e. ( W /. .~ ) ) -> ( K ` a ) = ( E ` a ) ) |
198 |
43 197
|
syldan |
|- ( ( ph /\ a e. X ) -> ( K ` a ) = ( E ` a ) ) |
199 |
17 21 198
|
eqfnfvd |
|- ( ph -> K = E ) |