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Theorem frgpcpbl 16583
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 2467 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 eqid 2467 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
4 eqid 2467 . . 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 2467 . . 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 2467 . . 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 16581 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A concat  B )  .~  ( C concat  D ) )
81, 2efger 16542 . . . . . 6  |-  .~  Er  (  _I  ` Word  ( I  X.  2o ) )
98a1i 11 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  .~  Er  (  _I  ` Word  ( I  X.  2o ) ) )
10 simpl 457 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  .~  C )
119, 10ercl 7322 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  (  _I  ` Word 
( I  X.  2o ) ) )
121efgrcl 16539 . . . . . . 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 463 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
1513simpld 459 . . . . . . 7  |-  ( ( A  .~  C  /\  B  .~  D )  ->  I  e.  _V )
16 2on 7138 . . . . . . 7  |-  2o  e.  On
17 xpexg 6586 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
1815, 16, 17sylancl 662 . . . . . 6  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( I  X.  2o )  e.  _V )
19 frgpval.b . . . . . . 7  |-  M  =  (freeMnd `  ( I  X.  2o ) )
20 eqid 2467 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2119, 20frmdbas 15852 . . . . . 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 2511 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  =  ( Base `  M
) )
2411, 23eleqtrd 2557 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  ( Base `  M ) )
25 simpr 461 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  .~  D )
269, 25ercl 7322 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  (  _I  ` Word 
( I  X.  2o ) ) )
2726, 23eleqtrd 2557 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  ( Base `  M ) )
28 frgpcpbl.p . . . 4  |-  .+  =  ( +g  `  M )
2919, 20, 28frmdadd 15855 . . 3  |-  ( ( A  e.  ( Base `  M )  /\  B  e.  ( Base `  M
) )  ->  ( A  .+  B )  =  ( A concat  B ) )
3024, 27, 29syl2anc 661 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  =  ( A concat  B ) )
319, 10ercl2 7324 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3231, 23eleqtrd 2557 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  ( Base `  M ) )
339, 25ercl2 7324 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3433, 23eleqtrd 2557 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  ( Base `  M ) )
3519, 20, 28frmdadd 15855 . . 3  |-  ( ( C  e.  ( Base `  M )  /\  D  e.  ( Base `  M
) )  ->  ( C  .+  D )  =  ( C concat  D ) )
3632, 34, 35syl2anc 661 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( C  .+  D
)  =  ( C concat  D ) )
377, 30, 363brtr4d 4477 1  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035   U_ciun 4325   class class class wbr 4447    |-> cmpt 4505    _I cid 4790   Oncon0 4878    X. cxp 4997   ran crn 5000   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1oc1o 7123   2oc2o 7124    Er wer 7308   0cc0 9492   1c1 9493    - cmin 9805   ...cfz 11672  ..^cfzo 11792   #chash 12373  Word cword 12500   concat cconcat 12502   splice csplice 12505   <"cs2 12769   Basecbs 14490   +g cplusg 14555  freeMndcfrmd 15847   ~FG cefg 16530  freeGrpcfrgp 16531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ec 7313  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-splice 12513  df-s2 12776  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-plusg 14568  df-frmd 15849  df-efg 16533
This theorem is referenced by:  frgp0  16584  frgpadd  16587
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