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Theorem gsumwrev 15879
Description: A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
gsumwrev.b  |-  B  =  ( Base `  M
)
gsumwrev.o  |-  O  =  (oppg
`  M )
Assertion
Ref Expression
gsumwrev  |-  ( ( M  e.  Mnd  /\  W  e. Word  B )  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) )

Proof of Theorem gsumwrev
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6097 . . . . 5  |-  ( x  =  (/)  ->  ( O 
gsumg  x )  =  ( O  gsumg  (/) ) )
2 fveq2 5689 . . . . . . 7  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (reverse `  (/) ) )
3 rev0 12402 . . . . . . 7  |-  (reverse `  (/) )  =  (/)
42, 3syl6eq 2489 . . . . . 6  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (/) )
54oveq2d 6105 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  (reverse `  x ) )  =  ( M  gsumg  (/) ) )
61, 5eqeq12d 2455 . . . 4  |-  ( x  =  (/)  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) ) )
76imbi2d 316 . . 3  |-  ( x  =  (/)  ->  ( ( M  e.  Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M 
gsumg  (/) ) ) ) )
8 oveq2 6097 . . . . 5  |-  ( x  =  y  ->  ( O  gsumg  x )  =  ( O  gsumg  y ) )
9 fveq2 5689 . . . . . 6  |-  ( x  =  y  ->  (reverse `  x )  =  (reverse `  y ) )
109oveq2d 6105 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  y ) ) )
118, 10eqeq12d 2455 . . . 4  |-  ( x  =  y  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y
) ) ) )
1211imbi2d 316 . . 3  |-  ( x  =  y  ->  (
( M  e.  Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  y )  =  ( M 
gsumg  (reverse `  y ) ) ) ) )
13 oveq2 6097 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( O  gsumg  x )  =  ( O  gsumg  ( y concat  <" z "> ) ) )
14 fveq2 5689 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  (reverse `  x )  =  (reverse `  ( y concat  <" z "> ) ) )
1514oveq2d 6105 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  ( y concat  <" z "> )
) ) )
1613, 15eqeq12d 2455 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x
) )  <->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) )
1716imbi2d 316 . . 3  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( M  e. 
Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x
) ) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) ) )
18 oveq2 6097 . . . . 5  |-  ( x  =  W  ->  ( O  gsumg  x )  =  ( O  gsumg  W ) )
19 fveq2 5689 . . . . . 6  |-  ( x  =  W  ->  (reverse `  x )  =  (reverse `  W ) )
2019oveq2d 6105 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  W ) ) )
2118, 20eqeq12d 2455 . . . 4  |-  ( x  =  W  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W
) ) ) )
2221imbi2d 316 . . 3  |-  ( x  =  W  ->  (
( M  e.  Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  W )  =  ( M 
gsumg  (reverse `  W ) ) ) ) )
23 gsumwrev.o . . . . . . 7  |-  O  =  (oppg
`  M )
24 eqid 2441 . . . . . . 7  |-  ( 0g
`  M )  =  ( 0g `  M
)
2523, 24oppgid 15869 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  O
)
2625gsum0 15508 . . . . 5  |-  ( O 
gsumg  (/) )  =  ( 0g
`  M )
2724gsum0 15508 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
2826, 27eqtr4i 2464 . . . 4  |-  ( O 
gsumg  (/) )  =  ( M 
gsumg  (/) )
2928a1i 11 . . 3  |-  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) )
30 oveq2 6097 . . . . . 6  |-  ( ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y )
)  ->  ( z
( +g  `  M ) ( O  gsumg  y ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
3123oppgmnd 15867 . . . . . . . . . 10  |-  ( M  e.  Mnd  ->  O  e.  Mnd )
3231adantr 465 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  O  e.  Mnd )
33 simprl 755 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  y  e. Word  B )
34 simprr 756 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  z  e.  B )
3534s1cld 12292 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  <" z ">  e. Word  B )
36 gsumwrev.b . . . . . . . . . . 11  |-  B  =  ( Base `  M
)
3723, 36oppgbas 15864 . . . . . . . . . 10  |-  B  =  ( Base `  O
)
38 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
3937, 38gsumccat 15517 . . . . . . . . 9  |-  ( ( O  e.  Mnd  /\  y  e. Word  B  /\  <" z ">  e. Word  B )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( ( O  gsumg  y ) ( +g  `  O
) ( O  gsumg  <" z "> ) ) )
4032, 33, 35, 39syl3anc 1218 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( ( O  gsumg  y ) ( +g  `  O
) ( O  gsumg  <" z "> ) ) )
4137gsumws1 15515 . . . . . . . . . . 11  |-  ( z  e.  B  ->  ( O  gsumg 
<" z "> )  =  z )
4241ad2antll 728 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg 
<" z "> )  =  z )
4342oveq2d 6105 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (
( O  gsumg  y ) ( +g  `  O ) ( O 
gsumg  <" z "> ) )  =  ( ( O  gsumg  y ) ( +g  `  O ) z ) )
44 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
4544, 23, 38oppgplus 15862 . . . . . . . . 9  |-  ( ( O  gsumg  y ) ( +g  `  O ) z )  =  ( z ( +g  `  M ) ( O  gsumg  y ) )
4643, 45syl6eq 2489 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (
( O  gsumg  y ) ( +g  `  O ) ( O 
gsumg  <" z "> ) )  =  ( z ( +g  `  M
) ( O  gsumg  y ) ) )
4740, 46eqtrd 2473 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( z ( +g  `  M ) ( O 
gsumg  y ) ) )
48 revccat 12404 . . . . . . . . . . 11  |-  ( ( y  e. Word  B  /\  <" z ">  e. Word  B )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
4933, 35, 48syl2anc 661 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
50 revs1 12403 . . . . . . . . . . 11  |-  (reverse `  <" z "> )  =  <" z ">
5150oveq1i 6099 . . . . . . . . . 10  |-  ( (reverse `  <" z "> ) concat  (reverse `  y
) )  =  (
<" z "> concat  (reverse `  y ) )
5249, 51syl6eq 2489 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( <" z "> concat  (reverse `  y )
) )
5352oveq2d 6105 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) )  =  ( M  gsumg  ( <" z "> concat  (reverse `  y )
) ) )
54 simpl 457 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  M  e.  Mnd )
55 revcl 12399 . . . . . . . . . 10  |-  ( y  e. Word  B  ->  (reverse `  y )  e. Word  B
)
5655ad2antrl 727 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  y )  e. Word  B
)
5736, 44gsumccat 15517 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  <" z ">  e. Word  B  /\  (reverse `  y
)  e. Word  B )  ->  ( M  gsumg  ( <" z "> concat  (reverse `  y )
) )  =  ( ( M  gsumg 
<" z "> ) ( +g  `  M
) ( M  gsumg  (reverse `  y
) ) ) )
5854, 35, 56, 57syl3anc 1218 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg  ( <" z "> concat  (reverse `  y )
) )  =  ( ( M  gsumg 
<" z "> ) ( +g  `  M
) ( M  gsumg  (reverse `  y
) ) ) )
5936gsumws1 15515 . . . . . . . . . 10  |-  ( z  e.  B  ->  ( M  gsumg 
<" z "> )  =  z )
6059ad2antll 728 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg 
<" z "> )  =  z )
6160oveq1d 6104 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (
( M  gsumg 
<" z "> ) ( +g  `  M
) ( M  gsumg  (reverse `  y
) ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
6253, 58, 613eqtrd 2477 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
6347, 62eqeq12d 2455 . . . . . 6  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (
( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) )  <->  ( z
( +g  `  M ) ( O  gsumg  y ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) ) )
6430, 63syl5ibr 221 . . . . 5  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (
( O  gsumg  y )  =  ( M  gsumg  (reverse `  y )
)  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) )
6564expcom 435 . . . 4  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( M  e.  Mnd  ->  ( ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) ) )
6665a2d 26 . . 3  |-  ( ( y  e. Word  B  /\  z  e.  B )  ->  ( ( M  e. 
Mnd  ->  ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y
) ) )  -> 
( M  e.  Mnd  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) ) )
677, 12, 17, 22, 29, 66wrdind 12369 . 2  |-  ( W  e. Word  B  ->  ( M  e.  Mnd  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) ) )
6867impcom 430 1  |-  ( ( M  e.  Mnd  /\  W  e. Word  B )  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   (/)c0 3635   ` cfv 5416  (class class class)co 6089  Word cword 12219   concat cconcat 12221   <"cs1 12222  reversecreverse 12225   Basecbs 14172   +g cplusg 14236   0gc0g 14376    gsumg cgsu 14377   Mndcmnd 15407  oppgcoppg 15858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  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-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-recs 6830  df-rdg 6864  df-1o 6918  df-oadd 6922  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-seq 11805  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-substr 12231  df-reverse 12233  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-gsum 14379  df-mnd 15413  df-submnd 15463  df-oppg 15859
This theorem is referenced by:  symgtrinv  15976
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