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Theorem frgpcpbl 16249
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 2441 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 eqid 2441 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
4 eqid 2441 . . 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 2441 . . 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 2441 . . 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 16247 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A concat  B )  .~  ( C concat  D ) )
81, 2efger 16208 . . . . . 6  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
98a1i 11 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) )
10 simpl 454 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  .~  C )
119, 10ercl 7108 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  (  _I  ` Word 
( I  X.  2o ) ) )
121efgrcl 16205 . . . . . . 7  |-  ( A  e.  (  _I  ` Word  ( I  X.  2o ) )  ->  (
I  e.  _V  /\  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) ) )
1311, 12syl 16 . . . . . 6  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( I  e.  _V  /\  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) ) )
1413simprd 460 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
1513simpld 456 . . . . . . 7  |-  ( ( A  .~  C  /\  B  .~  D )  ->  I  e.  _V )
16 2on 6924 . . . . . . 7  |-  2o  e.  On
17 xpexg 6506 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
1815, 16, 17sylancl 657 . . . . . 6  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( I  X.  2o )  e.  _V )
19 frgpval.b . . . . . . 7  |-  M  =  (freeMnd `  ( I  X.  2o ) )
20 eqid 2441 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2119, 20frmdbas 15523 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  M )  = Word  ( I  X.  2o ) )
2218, 21syl 16 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( Base `  M )  = Word  ( I  X.  2o ) )
2314, 22eqtr4d 2476 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  =  ( Base `  M
) )
2411, 23eleqtrd 2517 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  ( Base `  M ) )
25 simpr 458 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  .~  D )
269, 25ercl 7108 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  (  _I  ` Word 
( I  X.  2o ) ) )
2726, 23eleqtrd 2517 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  ( Base `  M ) )
28 frgpcpbl.p . . . 4  |-  .+  =  ( +g  `  M )
2919, 20, 28frmdadd 15526 . . 3  |-  ( ( A  e.  ( Base `  M )  /\  B  e.  ( Base `  M
) )  ->  ( A  .+  B )  =  ( A concat  B ) )
3024, 27, 29syl2anc 656 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  =  ( A concat  B ) )
319, 10ercl2 7110 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3231, 23eleqtrd 2517 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  ( Base `  M ) )
339, 25ercl2 7110 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3433, 23eleqtrd 2517 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  ( Base `  M ) )
3519, 20, 28frmdadd 15526 . . 3  |-  ( ( C  e.  ( Base `  M )  /\  D  e.  ( Base `  M
) )  ->  ( C  .+  D )  =  ( C concat  D ) )
3632, 34, 35syl2anc 656 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( C  .+  D
)  =  ( C concat  D ) )
377, 30, 363brtr4d 4319 1  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   A.wral 2713   {crab 2717   _Vcvv 2970    \ cdif 3322   (/)c0 3634   {csn 3874   <.cop 3880   <.cotp 3882   U_ciun 4168   class class class wbr 4289    e. cmpt 4347    _I cid 4627   Oncon0 4715    X. cxp 4834   ran crn 4837   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092   1oc1o 6909   2oc2o 6910    Er wer 7094   0cc0 9278   1c1 9279    - cmin 9591   ...cfz 11433  ..^cfzo 11544   #chash 12099  Word cword 12217   concat cconcat 12219   splice csplice 12222   <"cs2 12464   Basecbs 14170   +g cplusg 14234  freeMndcfrmd 15518   ~FG cefg 16196  freeGrpcfrgp 16197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-ot 3883  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-fzo 11545  df-hash 12100  df-word 12225  df-concat 12227  df-s1 12228  df-substr 12229  df-splice 12230  df-s2 12471  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-plusg 14247  df-frmd 15520  df-efg 16199
This theorem is referenced by:  frgp0  16250  frgpadd  16253
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