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Theorem frgpcpbl 15068
Description: Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
frgpval.m  |-  G  =  (freeGrp `  I )
frgpval.b  |-  M  =  (freeMnd `  ( I  X.  2o ) )
frgpval.r  |-  .~  =  ( ~FG  `  I )
frgpcpbl.p  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
frgpcpbl  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )

Proof of Theorem frgpcpbl
Dummy variables  k  m  n  t  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 eqid 2283 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
4 eqid 2283 . . 3  |-  ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) )  =  ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) )
5 eqid 2283 . . 3  |-  ( (  _I  ` Word  ( I  X.  2o ) )  \  U_ x  e.  (  _I  ` Word  ( I  X.  2o ) ) ran  (
( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  x ) )  =  ( (  _I  ` Word  ( I  X.  2o ) )  \  U_ x  e.  (  _I  ` Word 
( I  X.  2o ) ) ran  (
( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  x ) )
6 eqid 2283 . . 3  |-  ( m  e.  { t  e.  (Word  (  _I  ` Word  ( I  X.  2o ) )  \  { (/)
} )  |  ( ( t `  0
)  e.  ( (  _I  ` Word  ( I  X.  2o ) )  \  U_ x  e.  (  _I  ` Word  ( I  X.  2o ) ) ran  (
( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  x ) )  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )  =  ( m  e.  { t  e.  (Word  (  _I  ` Word  ( I  X.  2o ) )  \  { (/)
} )  |  ( ( t `  0
)  e.  ( (  _I  ` Word  ( I  X.  2o ) )  \  U_ x  e.  (  _I  ` Word  ( I  X.  2o ) ) ran  (
( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  x ) )  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( ( v  e.  (  _I  ` Word  ( I  X.  2o ) )  |->  ( n  e.  ( 0 ... ( # `  v
) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. ) `  w ) "> >.
) ) ) `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
71, 2, 3, 4, 5, 6efgcpbl2 15066 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A concat  B )  .~  ( C concat  D ) )
81, 2efger 15027 . . . . . 6  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
98a1i 10 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) )
10 simpl 443 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  .~  C )
119, 10ercl 6671 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  (  _I  ` Word 
( I  X.  2o ) ) )
121efgrcl 15024 . . . . . . 7  |-  ( A  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  (
I  e.  _V  /\  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) ) )
1311, 12syl 15 . . . . . 6  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( I  e.  _V  /\  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) ) )
1413simprd 449 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
1513simpld 445 . . . . . . 7  |-  ( ( A  .~  C  /\  B  .~  D )  ->  I  e.  _V )
16 2on 6487 . . . . . . 7  |-  2o  e.  On
17 xpexg 4800 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
1815, 16, 17sylancl 643 . . . . . 6  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( I  X.  2o )  e.  _V )
19 frgpval.b . . . . . . 7  |-  M  =  (freeMnd `  ( I  X.  2o ) )
20 eqid 2283 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2119, 20frmdbas 14474 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  M )  = Word  ( I  X.  2o ) )
2218, 21syl 15 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( Base `  M )  = Word  ( I  X.  2o ) )
2314, 22eqtr4d 2318 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  =  ( Base `  M
) )
2411, 23eleqtrd 2359 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  ( Base `  M ) )
25 simpr 447 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  .~  D )
269, 25ercl 6671 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  (  _I  ` Word 
( I  X.  2o ) ) )
2726, 23eleqtrd 2359 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  ( Base `  M ) )
28 frgpcpbl.p . . . 4  |-  .+  =  ( +g  `  M )
2919, 20, 28frmdadd 14477 . . 3  |-  ( ( A  e.  ( Base `  M )  /\  B  e.  ( Base `  M
) )  ->  ( A  .+  B )  =  ( A concat  B ) )
3024, 27, 29syl2anc 642 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  =  ( A concat  B ) )
319, 10ercl2 6673 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3231, 23eleqtrd 2359 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  ( Base `  M ) )
339, 25ercl2 6673 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3433, 23eleqtrd 2359 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  ( Base `  M ) )
3519, 20, 28frmdadd 14477 . . 3  |-  ( ( C  e.  ( Base `  M )  /\  D  e.  ( Base `  M
) )  ->  ( C  .+  D )  =  ( C concat  D ) )
3632, 34, 35syl2anc 642 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( C  .+  D
)  =  ( C concat  D ) )
377, 30, 363brtr4d 4053 1  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023    e. cmpt 4077    _I cid 4304   Oncon0 4392    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473    Er wer 6657   0cc0 8737   1c1 8738    - cmin 9037   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   concat cconcat 11404   splice csplice 11407   <"cs2 11491   Basecbs 13148   +g cplusg 13208  freeMndcfrmd 14469   ~FG cefg 15015  freeGrpcfrgp 15016
This theorem is referenced by:  frgp0  15069  frgpadd  15072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-frmd 14471  df-efg 15018
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