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Theorem frgpadd 15088
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 443 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  W )
2 simpr 447 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  W )
3 frgpadd.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
43efgrcl 15040 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
54adantr 451 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
65simpld 445 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  I  e.  _V )
7 frgpadd.g . . . . . 6  |-  G  =  (freeGrp `  I )
8 eqid 2296 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
9 frgpadd.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
107, 8, 9frgpval 15083 . . . . 5  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
116, 10syl 15 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
125simprd 449 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  = Word  ( I  X.  2o ) )
13 2on 6503 . . . . . . 7  |-  2o  e.  On
14 xpexg 4816 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
156, 13, 14sylancl 643 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  X.  2o )  e.  _V )
16 eqid 2296 . . . . . . 7  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
178, 16frmdbas 14490 . . . . . 6  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
1815, 17syl 15 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
1912, 18eqtr4d 2331 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
203, 9efger 15043 . . . . 5  |-  .~  Er  W
2120a1i 10 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  Er  W )
228frmdmnd 14497 . . . . 5  |-  ( ( I  X.  2o )  e.  _V  ->  (freeMnd `  ( I  X.  2o ) )  e.  Mnd )
2315, 22syl 15 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  (freeMnd `  ( I  X.  2o ) )  e. 
Mnd )
24 eqid 2296 . . . . . 6  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
257, 8, 9, 24frgpcpbl 15084 . . . . 5  |-  ( ( a  .~  b  /\  c  .~  d )  -> 
( a ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) c )  .~  (
b ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) d ) )
2625a1i 10 . . . 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 451 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
(freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28 simprl 732 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  W )
2919adantr 451 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3028, 29eleqtrd 2372 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 simprr 733 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  W )
3231, 29eleqtrd 2372 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3316, 24mndcl 14388 . . . . . 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 1182 . . . . 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 2373 . . . 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 13471 . . 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 1279 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  )
391, 19eleqtrd 2372 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
402, 19eleqtrd 2372 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
418, 16, 24frmdadd 14493 . . . 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 642 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B ) )
43 eceq1 6712 . . 3  |-  ( ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B )  ->  [ ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  =  [ ( A concat  B
) ]  .~  )
4442, 43syl 15 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  [ ( A ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) B ) ]  .~  =  [
( A concat  B ) ]  .~  )
4538, 44eqtrd 2328 1  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A concat  B ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   class class class wbr 4039    _I cid 4320   Oncon0 4408    X. cxp 4703   ` cfv 5271  (class class class)co 5874   2oc2o 6489    Er wer 6673   [cec 6674  Word cword 11419   concat cconcat 11420   Basecbs 13164   +g cplusg 13224    /.s cqus 13424   Mndcmnd 14377  freeMndcfrmd 14485   ~FG cefg 15031  freeGrpcfrgp 15032
This theorem is referenced by:  frgpinv  15089  frgpmhm  15090  frgpup1  15100  frgpnabllem1  15177
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-qs 6682  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  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-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  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-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-imas 13427  df-divs 13428  df-mnd 14383  df-frmd 14487  df-efg 15034  df-frgp 15035
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