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Theorem frgpadd 16655
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 457 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  W )
2 simpr 461 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  W )
3 frgpadd.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
43efgrcl 16607 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
54adantr 465 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
65simpld 459 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  I  e.  _V )
7 frgpadd.g . . . . . 6  |-  G  =  (freeGrp `  I )
8 eqid 2443 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
9 frgpadd.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
107, 8, 9frgpval 16650 . . . . 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 463 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  = Word  ( I  X.  2o ) )
13 2on 7140 . . . . . . 7  |-  2o  e.  On
14 xpexg 6587 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
156, 13, 14sylancl 662 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  X.  2o )  e.  _V )
16 eqid 2443 . . . . . . 7  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
178, 16frmdbas 15894 . . . . . 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 2487 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
203, 9efger 16610 . . . . 5  |-  .~  Er  W
2120a1i 11 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  Er  W )
228frmdmnd 15901 . . . . 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 2443 . . . . . 6  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
257, 8, 9, 24frgpcpbl 16651 . . . . 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 465 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
(freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28 simprl 756 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  W )
2919adantr 465 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3028, 29eleqtrd 2533 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 simprr 757 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  W )
3231, 29eleqtrd 2533 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3316, 24mndcl 15803 . . . . . 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 1229 . . . . 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 2534 . . . 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, 36qusaddval 14827 . . 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 1327 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  )
391, 19eleqtrd 2533 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
402, 19eleqtrd 2533 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
418, 16, 24frmdadd 15897 . . . 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 661 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B ) )
4342eceq1d 7350 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  [ ( A ( +g  `  (freeMnd `  (
I  X.  2o ) ) ) B ) ]  .~  =  [
( A concat  B ) ]  .~  )
4438, 43eqtrd 2484 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 1383    e. wcel 1804   _Vcvv 3095   class class class wbr 4437    _I cid 4780   Oncon0 4868    X. cxp 4987   ` cfv 5578  (class class class)co 6281   2oc2o 7126    Er wer 7310   [cec 7311  Word cword 12513   concat cconcat 12515   Basecbs 14509   +g cplusg 14574    /.s cqus 14779   Mndcmnd 15793  freeMndcfrmd 15889   ~FG cefg 16598  freeGrpcfrgp 16599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-substr 12525  df-splice 12526  df-s2 12792  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-ip 14592  df-tset 14593  df-ple 14594  df-ds 14596  df-imas 14782  df-qus 14783  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-frmd 15891  df-efg 16601  df-frgp 16602
This theorem is referenced by:  frgpinv  16656  frgpmhm  16657  frgpup1  16667  frgpnabllem1  16751
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