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Theorem frgpadd 15072
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 15024 . . . . . . 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 2283 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
9 frgpadd.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
107, 8, 9frgpval 15067 . . . . 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 6487 . . . . . . 7  |-  2o  e.  On
14 xpexg 4800 . . . . . . 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 2283 . . . . . . 7  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
178, 16frmdbas 14474 . . . . . 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 2318 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
203, 9efger 15027 . . . . 5  |-  .~  Er  W
2120a1i 10 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  Er  W )
228frmdmnd 14481 . . . . 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 2283 . . . . . 6  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
257, 8, 9, 24frgpcpbl 15068 . . . . 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 2359 . . . . . 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 2359 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3316, 24mndcl 14372 . . . . . 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 2360 . . . 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 13455 . . 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 2359 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
402, 19eleqtrd 2359 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
418, 16, 24frmdadd 14477 . . . 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 6696 . . 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 2315 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 1623    e. wcel 1684   _Vcvv 2788   class class class wbr 4023    _I cid 4304   Oncon0 4392    X. cxp 4687   ` cfv 5255  (class class class)co 5858   2oc2o 6473    Er wer 6657   [cec 6658  Word cword 11403   concat cconcat 11404   Basecbs 13148   +g cplusg 13208    /.s cqus 13408   Mndcmnd 14361  freeMndcfrmd 14469   ~FG cefg 15015  freeGrpcfrgp 15016
This theorem is referenced by:  frgpinv  15073  frgpmhm  15074  frgpup1  15084  frgpnabllem1  15161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-ot 3650  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-splice 11413  df-s2 11498  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-imas 13411  df-divs 13412  df-mnd 14367  df-frmd 14471  df-efg 15018  df-frgp 15019
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