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Theorem frgpnabllem2 15487
Description: Lemma for frgpnabl 15488. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
frgpnabl.g  |-  G  =  (freeGrp `  I )
frgpnabl.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpnabl.r  |-  .~  =  ( ~FG  `  I )
frgpnabl.p  |-  .+  =  ( +g  `  G )
frgpnabl.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
frgpnabl.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
frgpnabl.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
frgpnabl.u  |-  U  =  (varFGrp `  I )
frgpnabl.i  |-  ( ph  ->  I  e.  _V )
frgpnabl.a  |-  ( ph  ->  A  e.  I )
frgpnabl.b  |-  ( ph  ->  B  e.  I )
frgpnabl.n  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
Assertion
Ref Expression
frgpnabllem2  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    v, n, w, x, y, z, I    ph, x    x, 
.~ , y, z    x, B    n, W, v, w, x, y, z    x, G    n, M, v, w, x    x, T
Allowed substitution hints:    ph( y, z, w, v, n)    A( y, z, w, v, n)    B( y, z, w, v, n)    D( x, y, z, w, v, n)    .+ ( x, y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    U( x, y, z, w, v, n)    G( y,
z, w, v, n)    M( y, z)

Proof of Theorem frgpnabllem2
Dummy variables  d  m  t  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpnabl.a . 2  |-  ( ph  ->  A  e.  I )
2 0ex 4341 . . 3  |-  (/)  e.  _V
32a1i 11 . 2  |-  ( ph  -> 
(/)  e.  _V )
4 frgpnabl.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
5 difss 3476 . . . . . . . 8  |-  ( W 
\  U_ x  e.  W  ran  ( T `  x
) )  C_  W
64, 5eqsstri 3380 . . . . . . 7  |-  D  C_  W
7 inss1 3563 . . . . . . . 8  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  D
8 frgpnabl.g . . . . . . . . 9  |-  G  =  (freeGrp `  I )
9 frgpnabl.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
10 frgpnabl.r . . . . . . . . 9  |-  .~  =  ( ~FG  `  I )
11 frgpnabl.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
12 frgpnabl.m . . . . . . . . 9  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
13 frgpnabl.t . . . . . . . . 9  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
14 frgpnabl.u . . . . . . . . 9  |-  U  =  (varFGrp `  I )
15 frgpnabl.i . . . . . . . . 9  |-  ( ph  ->  I  e.  _V )
16 frgpnabl.b . . . . . . . . 9  |-  ( ph  ->  B  e.  I )
178, 9, 10, 11, 12, 13, 4, 14, 15, 16, 1frgpnabllem1 15486 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) ) )
187, 17sseldi 3348 . . . . . . 7  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  D )
196, 18sseldi 3348 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  W )
20 eqid 2438 . . . . . . 7  |-  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )  =  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
219, 10, 12, 13, 4, 20efgredeu 15386 . . . . . 6  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  W  ->  E! d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
22 reurmo 2925 . . . . . 6  |-  ( E! d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  E* d  e.  D d  .~  <"
<. B ,  (/) >. <. A ,  (/)
>. "> )
2319, 21, 223syl 19 . . . . 5  |-  ( ph  ->  E* d  e.  D
d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
24 inss1 3563 . . . . . 6  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  D
258, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16frgpnabllem1 15486 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
2624, 25sseldi 3348 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
279, 10efger 15352 . . . . . . . . 9  |-  .~  Er  W
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  .~  Er  W )
298frgpgrp 15396 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  e.  Grp )
3015, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
31 eqid 2438 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3210, 14, 8, 31vrgpf 15402 . . . . . . . . . . . 12  |-  ( I  e.  _V  ->  U : I --> ( Base `  G ) )
3315, 32syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U : I --> ( Base `  G ) )
3433, 1ffvelrnd 5873 . . . . . . . . . 10  |-  ( ph  ->  ( U `  A
)  e.  ( Base `  G ) )
3533, 16ffvelrnd 5873 . . . . . . . . . 10  |-  ( ph  ->  ( U `  B
)  e.  ( Base `  G ) )
3631, 11grpcl 14820 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( U `  A )  e.  ( Base `  G
)  /\  ( U `  B )  e.  (
Base `  G )
)  ->  ( ( U `  A )  .+  ( U `  B
) )  e.  (
Base `  G )
)
3730, 34, 35, 36syl3anc 1185 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( Base `  G ) )
38 eqid 2438 . . . . . . . . . . . 12  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
398, 38, 10frgpval 15392 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
4015, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
41 2on 6734 . . . . . . . . . . . . . 14  |-  2o  e.  On
42 xpexg 4991 . . . . . . . . . . . . . 14  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4315, 41, 42sylancl 645 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
44 wrdexg 11741 . . . . . . . . . . . . 13  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
45 fvi 5785 . . . . . . . . . . . . 13  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4643, 44, 453syl 19 . . . . . . . . . . . 12  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
479, 46syl5eq 2482 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
48 eqid 2438 . . . . . . . . . . . . 13  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4938, 48frmdbas 14799 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
5043, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
5147, 50eqtr4d 2473 . . . . . . . . . 10  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
52 fvex 5744 . . . . . . . . . . . 12  |-  ( ~FG  `  I
)  e.  _V
5310, 52eqeltri 2508 . . . . . . . . . . 11  |-  .~  e.  _V
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .~  e.  _V )
55 fvex 5744 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5655a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5740, 51, 54, 56divsbas 13772 . . . . . . . . 9  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
5837, 57eleqtrrd 2515 . . . . . . . 8  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  ) )
59 inss2 3564 . . . . . . . . 9  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  (
( U `  A
)  .+  ( U `  B ) )
6059, 25sseldi 3348 . . . . . . . 8  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
61 qsel 6985 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  )
6228, 58, 60, 61syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. A ,  (/) >. <. B ,  (/)
>. "> ]  .~  )
63 inss2 3564 . . . . . . . . . 10  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  (
( U `  B
)  .+  ( U `  A ) )
6463, 17sseldi 3348 . . . . . . . . 9  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  B ) 
.+  ( U `  A ) ) )
65 frgpnabl.n . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
6664, 65eleqtrrd 2515 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
67 qsel 6985 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. B ,  (/) >. <. A ,  (/) >. "> ]  .~  )
6828, 58, 66, 67syl3anc 1185 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. B ,  (/) >. <. A ,  (/)
>. "> ]  .~  )
6962, 68eqtr3d 2472 . . . . . 6  |-  ( ph  ->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
706, 26sseldi 3348 . . . . . . 7  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
7128, 70erth 6951 . . . . . 6  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  <->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
)
7269, 71mpbird 225 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7328, 19erref 6927 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
74 breq1 4217 . . . . . 6  |-  ( d  =  <" <. A ,  (/)
>. <. B ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
75 breq1 4217 . . . . . 6  |-  ( d  =  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
7674, 75rmoi 3252 . . . . 5  |-  ( ( E* d  e.  D
d  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  /\  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  D  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )  /\  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  D  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )  ->  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" <. B ,  (/)
>. <. A ,  (/) >. "> )
7723, 26, 72, 18, 73, 76syl122anc 1194 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7877fveq1d 5732 . . 3  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )
)
79 opex 4429 . . . 4  |-  <. A ,  (/)
>.  e.  _V
80 s2fv0 11851 . . . 4  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
8179, 80ax-mp 8 . . 3  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
82 opex 4429 . . . 4  |-  <. B ,  (/)
>.  e.  _V
83 s2fv0 11851 . . . 4  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
)
8482, 83ax-mp 8 . . 3  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
8578, 81, 843eqtr3g 2493 . 2  |-  ( ph  -> 
<. A ,  (/) >.  =  <. B ,  (/) >. )
86 opthg 4438 . . 3  |-  ( ( A  e.  I  /\  (/) 
e.  _V )  ->  ( <. A ,  (/) >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  (/)  =  (/) ) ) )
8786simprbda 608 . 2  |-  ( ( ( A  e.  I  /\  (/)  e.  _V )  /\  <. A ,  (/) >.  =  <. B ,  (/) >.
)  ->  A  =  B )
881, 3, 85, 87syl21anc 1184 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   E!wreu 2709   E*wrmo 2710   {crab 2711   _Vcvv 2958    \ cdif 3319    i^i cin 3321   (/)c0 3630   {csn 3816   <.cop 3819   <.cotp 3820   U_ciun 4095   class class class wbr 4214    e. cmpt 4268    _I cid 4495   Oncon0 4583    X. cxp 4878   ran crn 4881   -->wf 5452   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   1oc1o 6719   2oc2o 6720    Er wer 6904   [cec 6905   /.cqs 6906   0cc0 8992   1c1 8993    - cmin 9293   ...cfz 11045  ..^cfzo 11137   #chash 11620  Word cword 11719   splice csplice 11723   <"cs2 11807   Basecbs 13471   +g cplusg 13531    /.s cqus 13733   Grpcgrp 14687  freeMndcfrmd 14794   ~FG cefg 15340  freeGrpcfrgp 15341  varFGrpcvrgp 15342
This theorem is referenced by:  frgpnabl  15488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-oadd 6730  df-er 6907  df-ec 6909  df-qs 6913  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-splice 11729  df-reverse 11730  df-s2 11814  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-0g 13729  df-imas 13736  df-divs 13737  df-mnd 14692  df-frmd 14796  df-grp 14814  df-efg 15343  df-frgp 15344  df-vrgp 15345
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