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Theorem gsumwrev 16206
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 6292 . . . . 5  |-  ( x  =  (/)  ->  ( O 
gsumg  x )  =  ( O  gsumg  (/) ) )
2 fveq2 5866 . . . . . . 7  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (reverse `  (/) ) )
3 rev0 12701 . . . . . . 7  |-  (reverse `  (/) )  =  (/)
42, 3syl6eq 2524 . . . . . 6  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (/) )
54oveq2d 6300 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  (reverse `  x ) )  =  ( M  gsumg  (/) ) )
61, 5eqeq12d 2489 . . . 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 6292 . . . . 5  |-  ( x  =  y  ->  ( O  gsumg  x )  =  ( O  gsumg  y ) )
9 fveq2 5866 . . . . . 6  |-  ( x  =  y  ->  (reverse `  x )  =  (reverse `  y ) )
109oveq2d 6300 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  y ) ) )
118, 10eqeq12d 2489 . . . 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 6292 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( O  gsumg  x )  =  ( O  gsumg  ( y concat  <" z "> ) ) )
14 fveq2 5866 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  (reverse `  x )  =  (reverse `  ( y concat  <" z "> ) ) )
1514oveq2d 6300 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  ( y concat  <" z "> )
) ) )
1613, 15eqeq12d 2489 . . . 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 6292 . . . . 5  |-  ( x  =  W  ->  ( O  gsumg  x )  =  ( O  gsumg  W ) )
19 fveq2 5866 . . . . . 6  |-  ( x  =  W  ->  (reverse `  x )  =  (reverse `  W ) )
2019oveq2d 6300 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  W ) ) )
2118, 20eqeq12d 2489 . . . 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 2467 . . . . . . 7  |-  ( 0g
`  M )  =  ( 0g `  M
)
2523, 24oppgid 16196 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  O
)
2625gsum0 15832 . . . . 5  |-  ( O 
gsumg  (/) )  =  ( 0g
`  M )
2724gsum0 15832 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
2826, 27eqtr4i 2499 . . . 4  |-  ( O 
gsumg  (/) )  =  ( M 
gsumg  (/) )
2928a1i 11 . . 3  |-  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) )
30 oveq2 6292 . . . . . 6  |-  ( ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y )
)  ->  ( z
( +g  `  M ) ( O  gsumg  y ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
3123oppgmnd 16194 . . . . . . . . . 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 12578 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  <" z ">  e. Word  B )
36 gsumwrev.b . . . . . . . . . . 11  |-  B  =  ( Base `  M
)
3723, 36oppgbas 16191 . . . . . . . . . 10  |-  B  =  ( Base `  O
)
38 eqid 2467 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
3937, 38gsumccat 15841 . . . . . . . . 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 1228 . . . . . . . 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 15839 . . . . . . . . . . 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 6300 . . . . . . . . 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 2467 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
4544, 23, 38oppgplus 16189 . . . . . . . . 9  |-  ( ( O  gsumg  y ) ( +g  `  O ) z )  =  ( z ( +g  `  M ) ( O  gsumg  y ) )
4643, 45syl6eq 2524 . . . . . . . 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 2508 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( z ( +g  `  M ) ( O 
gsumg  y ) ) )
48 revccat 12703 . . . . . . . . . . 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 12702 . . . . . . . . . . 11  |-  (reverse `  <" z "> )  =  <" z ">
5150oveq1i 6294 . . . . . . . . . 10  |-  ( (reverse `  <" z "> ) concat  (reverse `  y
) )  =  (
<" z "> concat  (reverse `  y ) )
5249, 51syl6eq 2524 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( <" z "> concat  (reverse `  y )
) )
5352oveq2d 6300 . . . . . . . 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 12698 . . . . . . . . . 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 15841 . . . . . . . . 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 1228 . . . . . . . 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 15839 . . . . . . . . . 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 6299 . . . . . . . 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 2512 . . . . . . 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 2489 . . . . . 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 12665 . 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 1379    e. wcel 1767   (/)c0 3785   ` cfv 5588  (class class class)co 6284  Word cword 12500   concat cconcat 12502   <"cs1 12503  reversecreverse 12506   Basecbs 14490   +g cplusg 14555   0gc0g 14695    gsumg cgsu 14696   Mndcmnd 15726  oppgcoppg 16185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-seq 12076  df-hash 12374  df-word 12508  df-concat 12510  df-s1 12511  df-substr 12512  df-reverse 12514  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-0g 14697  df-gsum 14698  df-mnd 15732  df-submnd 15787  df-oppg 16186
This theorem is referenced by:  symgtrinv  16303
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