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Theorem frgpadd 15395
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 444 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  W )
2 simpr 448 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  W )
3 frgpadd.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
43efgrcl 15347 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
54adantr 452 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
65simpld 446 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  I  e.  _V )
7 frgpadd.g . . . . . 6  |-  G  =  (freeGrp `  I )
8 eqid 2436 . . . . . 6  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
9 frgpadd.r . . . . . 6  |-  .~  =  ( ~FG  `  I )
107, 8, 9frgpval 15390 . . . . 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 450 . . . . 5  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  = Word  ( I  X.  2o ) )
13 2on 6732 . . . . . . 7  |-  2o  e.  On
14 xpexg 4989 . . . . . . 7  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
156, 13, 14sylancl 644 . . . . . 6  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( I  X.  2o )  e.  _V )
16 eqid 2436 . . . . . . 7  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
178, 16frmdbas 14797 . . . . . 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 2471 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
203, 9efger 15350 . . . . 5  |-  .~  Er  W
2120a1i 11 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  Er  W )
228frmdmnd 14804 . . . . 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 2436 . . . . . 6  |-  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )  =  ( +g  `  (freeMnd `  ( I  X.  2o ) ) )
257, 8, 9, 24frgpcpbl 15391 . . . . 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 452 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
(freeMnd `  ( I  X.  2o ) )  e.  Mnd )
28 simprl 733 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  W )
2919adantr 452 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3028, 29eleqtrd 2512 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
b  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
31 simprr 734 . . . . . . 7  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  W )
3231, 29eleqtrd 2512 . . . . . 6  |-  ( ( ( A  e.  W  /\  B  e.  W
)  /\  ( b  e.  W  /\  d  e.  W ) )  -> 
d  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
3316, 24mndcl 14695 . . . . . 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 1184 . . . . 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 2513 . . . 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 13778 . . 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 1281 . 2  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B ) ]  .~  )
391, 19eleqtrd 2512 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  A  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
402, 19eleqtrd 2512 . . . 4  |-  ( ( A  e.  W  /\  B  e.  W )  ->  B  e.  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
418, 16, 24frmdadd 14800 . . . 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 643 . . 3  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( A ( +g  `  (freeMnd `  ( I  X.  2o ) ) ) B )  =  ( A concat  B ) )
43 eceq1 6941 . . 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 2468 1  |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [
( A concat  B ) ]  .~  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   class class class wbr 4212    _I cid 4493   Oncon0 4581    X. cxp 4876   ` cfv 5454  (class class class)co 6081   2oc2o 6718    Er wer 6902   [cec 6903  Word cword 11717   concat cconcat 11718   Basecbs 13469   +g cplusg 13529    /.s cqus 13731   Mndcmnd 14684  freeMndcfrmd 14792   ~FG cefg 15338  freeGrpcfrgp 15339
This theorem is referenced by:  frgpinv  15396  frgpmhm  15397  frgpup1  15407  frgpnabllem1  15484
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-splice 11727  df-s2 11812  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-imas 13734  df-divs 13735  df-mnd 14690  df-frmd 14794  df-efg 15341  df-frgp 15342
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