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Theorem frgpcpbl 16256
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 2443 . . 3  |-  (  _I 
` Word  ( I  X.  2o ) )  =  (  _I  ` Word  ( I  X.  2o ) )
2 frgpval.r . . 3  |-  .~  =  ( ~FG  `  I )
3 eqid 2443 . . 3  |-  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \ 
z ) >. )  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z )
>. )
4 eqid 2443 . . 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 2443 . . 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 2443 . . 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 16254 . 2  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A concat  B )  .~  ( C concat  D ) )
81, 2efger 16215 . . . . . 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 7112 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  (  _I  ` Word 
( I  X.  2o ) ) )
121efgrcl 16212 . . . . . . 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 6928 . . . . . . 7  |-  2o  e.  On
17 xpexg 6507 . . . . . . 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 2443 . . . . . . 7  |-  ( Base `  M )  =  (
Base `  M )
2119, 20frmdbas 15530 . . . . . 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 2478 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
(  _I  ` Word  ( I  X.  2o ) )  =  ( Base `  M
) )
2411, 23eleqtrd 2519 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  A  e.  ( Base `  M ) )
25 simpr 461 . . . . 5  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  .~  D )
269, 25ercl 7112 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  (  _I  ` Word 
( I  X.  2o ) ) )
2726, 23eleqtrd 2519 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  B  e.  ( Base `  M ) )
28 frgpcpbl.p . . . 4  |-  .+  =  ( +g  `  M )
2919, 20, 28frmdadd 15533 . . 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 7114 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3231, 23eleqtrd 2519 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  C  e.  ( Base `  M ) )
339, 25ercl2 7114 . . . 4  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  (  _I  ` Word 
( I  X.  2o ) ) )
3433, 23eleqtrd 2519 . . 3  |-  ( ( A  .~  C  /\  B  .~  D )  ->  D  e.  ( Base `  M ) )
3519, 20, 28frmdadd 15533 . . 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 4322 1  |-  ( ( A  .~  C  /\  B  .~  D )  -> 
( A  .+  B
)  .~  ( C  .+  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   {crab 2719   _Vcvv 2972    \ cdif 3325   (/)c0 3637   {csn 3877   <.cop 3883   <.cotp 3885   U_ciun 4171   class class class wbr 4292    e. cmpt 4350    _I cid 4631   Oncon0 4719    X. cxp 4838   ran crn 4841   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1oc1o 6913   2oc2o 6914    Er wer 7098   0cc0 9282   1c1 9283    - cmin 9595   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   concat cconcat 12223   splice csplice 12226   <"cs2 12468   Basecbs 14174   +g cplusg 14238  freeMndcfrmd 15525   ~FG cefg 16203  freeGrpcfrgp 16204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-s2 12475  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-plusg 14251  df-frmd 15527  df-efg 16206
This theorem is referenced by:  frgp0  16257  frgpadd  16260
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