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Theorem frgpcpbl 15084
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 2296 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 eqid 2296 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
4 eqid 2296 . . 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 2296 . . 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 2296 . . 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 15082 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A concat  B )  .~  ( C concat  D ) )
81, 2efger 15043 . . . . . 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 6687 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  (  _I  ` Word 
( I  X.  2o ) ) )
121efgrcl 15040 . . . . . . 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 6503 . . . . . . 7  |-  2o  e.  On
17 xpexg 4816 . . . . . . 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 2296 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2119, 20frmdbas 14490 . . . . . 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 2331 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  =  ( Base `  M
) )
2411, 23eleqtrd 2372 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  ( Base `  M ) )
25 simpr 447 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  .~  D )
269, 25ercl 6687 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  (  _I  ` Word 
( I  X.  2o ) ) )
2726, 23eleqtrd 2372 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  ( Base `  M ) )
28 frgpcpbl.p . . . 4  |-  .+  =  ( +g  `  M )
2919, 20, 28frmdadd 14493 . . 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 6689 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3231, 23eleqtrd 2372 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  ( Base `  M ) )
339, 25ercl2 6689 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3433, 23eleqtrd 2372 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  ( Base `  M ) )
3519, 20, 28frmdadd 14493 . . 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 4069 1  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039    e. cmpt 4093    _I cid 4320   Oncon0 4408    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489    Er wer 6673   0cc0 8753   1c1 8754    - cmin 9053   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   concat cconcat 11420   splice csplice 11423   <"cs2 11507   Basecbs 13164   +g cplusg 13224  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgp0  15085  frgpadd  15088
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-ec 6678  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-concat 11426  df-s1 11427  df-substr 11428  df-splice 11429  df-s2 11514  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-frmd 14487  df-efg 15034
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