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Theorem frgpadd 16253
Description: Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
frgpadd.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpadd.g  |-  G  =  (freeGrp `  I )
frgpadd.r  |-  .~  =  ( ~FG  `  I )
frgpadd.n  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
frgpadd  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A concat  B ) ]  .~  )

Proof of Theorem frgpadd
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 454 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  W )
2 simpr 458 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  W )
3 frgpadd.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
43efgrcl 16205 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
54adantr 462 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
65simpld 456 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  I  e.  _V )
7 frgpadd.g . . . . . 6  |-  G  =  (freeGrp `  I )
8 eqid 2441 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
9 frgpadd.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
107, 8, 9frgpval 16248 . . . . 5  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
116, 10syl 16 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
125simprd 460 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  = Word  ( I  X.  2o ) )
13 2on 6924 . . . . . . 7  |-  2o  e.  On
14 xpexg 6506 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
156, 13, 14sylancl 657 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  X.  2o )  e.  _V )
16 eqid 2441 . . . . . . 7  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
178, 16frmdbas 15523 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
1815, 17syl 16 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
1912, 18eqtr4d 2476 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
203, 9efger 16208 . . . . 5  |-  .~  Er  W
2120a1i 11 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  Er  W )
228frmdmnd 15530 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
2315, 22syl 16 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  (freeMnd `  ( I  X.  2o ) )  e. 
Mnd )
24 eqid 2441 . . . . . 6  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
257, 8, 9, 24frgpcpbl 16249 . . . . 5  |-  ( ( a  .~  b  /\  c  .~  d )  -> 
( a ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) c )  .~  (
b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d ) )
2625a1i 11 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( ( a  .~  b  /\  c  .~  d
)  ->  ( a
( +g  `  (freeMnd `  (
I  X.  2o ) ) ) c )  .~  ( b ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) d ) ) )
2723adantr 462 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
(freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28 simprl 750 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  W )
2919adantr 462 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3028, 29eleqtrd 2517 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 simprr 751 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  W )
3231, 29eleqtrd 2517 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3316, 24mndcl 15416 . . . . . 6  |-  ( ( (freeMnd `  ( I  X.  2o ) )  e. 
Mnd  /\  b  e.  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  /\  d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )  ->  ( b ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) d )  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3427, 30, 32, 33syl3anc 1213 . . . . 5  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
( b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d )  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )
3534, 29eleqtrrd 2518 . . . 4  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
( b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d )  e.  W
)
36 frgpadd.n . . . 4  |-  .+  =  ( +g  `  G )
3711, 19, 21, 23, 26, 35, 24, 36divsaddval 14487 . . 3  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  A  e.  W  /\  B  e.  W
)  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [ ( A ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) B ) ]  .~  )
381, 2, 37mpd3an23 1311 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  )
391, 19eleqtrd 2517 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
402, 19eleqtrd 2517 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
418, 16, 24frmdadd 15526 . . . 4  |-  ( ( A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  /\  B  e.  (
Base `  (freeMnd `  (
I  X.  2o ) ) ) )  -> 
( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B ) )
4239, 40, 41syl2anc 656 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B ) )
43 eceq1 7133 . . 3  |-  ( ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B )  ->  [ ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  =  [ ( A concat  B
) ]  .~  )
4442, 43syl 16 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  [ ( A ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) B ) ]  .~  =  [
( A concat  B ) ]  .~  )
4538, 44eqtrd 2473 1  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A concat  B ) ]  .~  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970   class class class wbr 4289    _I cid 4627   Oncon0 4715    X. cxp 4834   ` cfv 5415  (class class class)co 6090   2oc2o 6910    Er wer 7094   [cec 7095  Word cword 12217   concat cconcat 12219   Basecbs 14170   +g cplusg 14234    /.s cqus 14439   Mndcmnd 15405  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-qs 7103  df-map 7212  df-pm 7213  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  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-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  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-mulr 14248  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-imas 14442  df-divs 14443  df-mnd 15411  df-frmd 15520  df-efg 16199  df-frgp 16200
This theorem is referenced by:  frgpinv  16254  frgpmhm  16255  frgpup1  16265  frgpnabllem1  16344
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