Metamath Proof Explorer


Theorem frgpup2

Description: The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015) (Revised by Mario Carneiro, 28-Feb-2016)

Ref Expression
Hypotheses frgpup.b
|- B = ( Base ` H )
frgpup.n
|- N = ( invg ` H )
frgpup.t
|- T = ( y e. I , z e. 2o |-> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) )
frgpup.h
|- ( ph -> H e. Grp )
frgpup.i
|- ( ph -> I e. V )
frgpup.a
|- ( ph -> F : I --> B )
frgpup.w
|- W = ( _I ` Word ( I X. 2o ) )
frgpup.r
|- .~ = ( ~FG ` I )
frgpup.g
|- G = ( freeGrp ` I )
frgpup.x
|- X = ( Base ` G )
frgpup.e
|- E = ran ( g e. W |-> <. [ g ] .~ , ( H gsum ( T o. g ) ) >. )
frgpup.u
|- U = ( varFGrp ` I )
frgpup.y
|- ( ph -> A e. I )
Assertion frgpup2
|- ( ph -> ( E ` ( U ` A ) ) = ( F ` A ) )

Proof

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 frgpup.y
 |-  ( ph -> A e. I )
14 8 12 vrgpval
 |-  ( ( I e. V /\ A e. I ) -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ )
15 5 13 14 syl2anc
 |-  ( ph -> ( U ` A ) = [ <" <. A , (/) >. "> ] .~ )
16 15 fveq2d
 |-  ( ph -> ( E ` ( U ` A ) ) = ( E ` [ <" <. A , (/) >. "> ] .~ ) )
17 0ex
 |-  (/) e. _V
18 17 prid1
 |-  (/) e. { (/) , 1o }
19 df2o3
 |-  2o = { (/) , 1o }
20 18 19 eleqtrri
 |-  (/) e. 2o
21 opelxpi
 |-  ( ( A e. I /\ (/) e. 2o ) -> <. A , (/) >. e. ( I X. 2o ) )
22 13 20 21 sylancl
 |-  ( ph -> <. A , (/) >. e. ( I X. 2o ) )
23 22 s1cld
 |-  ( ph -> <" <. A , (/) >. "> e. Word ( I X. 2o ) )
24 2on
 |-  2o e. On
25 xpexg
 |-  ( ( I e. V /\ 2o e. On ) -> ( I X. 2o ) e. _V )
26 5 24 25 sylancl
 |-  ( ph -> ( I X. 2o ) e. _V )
27 wrdexg
 |-  ( ( I X. 2o ) e. _V -> Word ( I X. 2o ) e. _V )
28 fvi
 |-  ( Word ( I X. 2o ) e. _V -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
29 26 27 28 3syl
 |-  ( ph -> ( _I ` Word ( I X. 2o ) ) = Word ( I X. 2o ) )
30 7 29 syl5eq
 |-  ( ph -> W = Word ( I X. 2o ) )
31 23 30 eleqtrrd
 |-  ( ph -> <" <. A , (/) >. "> e. W )
32 1 2 3 4 5 6 7 8 9 10 11 frgpupval
 |-  ( ( ph /\ <" <. A , (/) >. "> e. W ) -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsum ( T o. <" <. A , (/) >. "> ) ) )
33 31 32 mpdan
 |-  ( ph -> ( E ` [ <" <. A , (/) >. "> ] .~ ) = ( H gsum ( T o. <" <. A , (/) >. "> ) ) )
34 1 2 3 4 5 6 frgpuptf
 |-  ( ph -> T : ( I X. 2o ) --> B )
35 s1co
 |-  ( ( <. A , (/) >. e. ( I X. 2o ) /\ T : ( I X. 2o ) --> B ) -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> )
36 22 34 35 syl2anc
 |-  ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( T ` <. A , (/) >. ) "> )
37 df-ov
 |-  ( A T (/) ) = ( T ` <. A , (/) >. )
38 iftrue
 |-  ( z = (/) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` y ) )
39 fveq2
 |-  ( y = A -> ( F ` y ) = ( F ` A ) )
40 38 39 sylan9eqr
 |-  ( ( y = A /\ z = (/) ) -> if ( z = (/) , ( F ` y ) , ( N ` ( F ` y ) ) ) = ( F ` A ) )
41 fvex
 |-  ( F ` A ) e. _V
42 40 3 41 ovmpoa
 |-  ( ( A e. I /\ (/) e. 2o ) -> ( A T (/) ) = ( F ` A ) )
43 13 20 42 sylancl
 |-  ( ph -> ( A T (/) ) = ( F ` A ) )
44 37 43 eqtr3id
 |-  ( ph -> ( T ` <. A , (/) >. ) = ( F ` A ) )
45 44 s1eqd
 |-  ( ph -> <" ( T ` <. A , (/) >. ) "> = <" ( F ` A ) "> )
46 36 45 eqtrd
 |-  ( ph -> ( T o. <" <. A , (/) >. "> ) = <" ( F ` A ) "> )
47 46 oveq2d
 |-  ( ph -> ( H gsum ( T o. <" <. A , (/) >. "> ) ) = ( H gsum <" ( F ` A ) "> ) )
48 6 13 ffvelrnd
 |-  ( ph -> ( F ` A ) e. B )
49 1 gsumws1
 |-  ( ( F ` A ) e. B -> ( H gsum <" ( F ` A ) "> ) = ( F ` A ) )
50 48 49 syl
 |-  ( ph -> ( H gsum <" ( F ` A ) "> ) = ( F ` A ) )
51 47 50 eqtrd
 |-  ( ph -> ( H gsum ( T o. <" <. A , (/) >. "> ) ) = ( F ` A ) )
52 16 33 51 3eqtrd
 |-  ( ph -> ( E ` ( U ` A ) ) = ( F ` A ) )