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Theorem gsumwrev 15164
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 6091 . . . . 5  |-  ( x  =  (/)  ->  ( O 
gsumg  x )  =  ( O  gsumg  (/) ) )
2 fveq2 5730 . . . . . . 7  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (reverse `  (/) ) )
3 rev0 11798 . . . . . . 7  |-  (reverse `  (/) )  =  (/)
42, 3syl6eq 2486 . . . . . 6  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (/) )
54oveq2d 6099 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  (reverse `  x ) )  =  ( M  gsumg  (/) ) )
61, 5eqeq12d 2452 . . . 4  |-  ( x  =  (/)  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) ) )
76imbi2d 309 . . 3  |-  ( x  =  (/)  ->  ( ( M  e.  Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M 
gsumg  (/) ) ) ) )
8 oveq2 6091 . . . . 5  |-  ( x  =  y  ->  ( O  gsumg  x )  =  ( O  gsumg  y ) )
9 fveq2 5730 . . . . . 6  |-  ( x  =  y  ->  (reverse `  x )  =  (reverse `  y ) )
109oveq2d 6099 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  y ) ) )
118, 10eqeq12d 2452 . . . 4  |-  ( x  =  y  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y
) ) ) )
1211imbi2d 309 . . 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 6091 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( O  gsumg  x )  =  ( O  gsumg  ( y concat  <" z "> ) ) )
14 fveq2 5730 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  (reverse `  x )  =  (reverse `  ( y concat  <" z "> ) ) )
1514oveq2d 6099 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  ( y concat  <" z "> )
) ) )
1613, 15eqeq12d 2452 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x
) )  <->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) )
1716imbi2d 309 . . 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 6091 . . . . 5  |-  ( x  =  W  ->  ( O  gsumg  x )  =  ( O  gsumg  W ) )
19 fveq2 5730 . . . . . 6  |-  ( x  =  W  ->  (reverse `  x )  =  (reverse `  W ) )
2019oveq2d 6099 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  W ) ) )
2118, 20eqeq12d 2452 . . . 4  |-  ( x  =  W  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W
) ) ) )
2221imbi2d 309 . . 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 2438 . . . . . . 7  |-  ( 0g
`  M )  =  ( 0g `  M
)
2523, 24oppgid 15154 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  O
)
2625gsum0 14782 . . . . 5  |-  ( O 
gsumg  (/) )  =  ( 0g
`  M )
2724gsum0 14782 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
2826, 27eqtr4i 2461 . . . 4  |-  ( O 
gsumg  (/) )  =  ( M 
gsumg  (/) )
2928a1i 11 . . 3  |-  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) )
30 oveq2 6091 . . . . . 6  |-  ( ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y )
)  ->  ( z
( +g  `  M ) ( O  gsumg  y ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
3123oppgmnd 15152 . . . . . . . . . 10  |-  ( M  e.  Mnd  ->  O  e.  Mnd )
3231adantr 453 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  O  e.  Mnd )
33 simprl 734 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  y  e. Word  B )
34 simprr 735 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  z  e.  B )
3534s1cld 11758 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  <" z ">  e. Word  B )
36 gsumwrev.b . . . . . . . . . . 11  |-  B  =  ( Base `  M
)
3723, 36oppgbas 15149 . . . . . . . . . 10  |-  B  =  ( Base `  O
)
38 eqid 2438 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
3937, 38gsumccat 14789 . . . . . . . . 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 1185 . . . . . . . 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 14787 . . . . . . . . . . 11  |-  ( z  e.  B  ->  ( O  gsumg 
<" z "> )  =  z )
4241ad2antll 711 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg 
<" z "> )  =  z )
4342oveq2d 6099 . . . . . . . . 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 2438 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
4544, 23, 38oppgplus 15147 . . . . . . . . 9  |-  ( ( O  gsumg  y ) ( +g  `  O ) z )  =  ( z ( +g  `  M ) ( O  gsumg  y ) )
4643, 45syl6eq 2486 . . . . . . . 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 2470 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( z ( +g  `  M ) ( O 
gsumg  y ) ) )
48 revccat 11800 . . . . . . . . . . 11  |-  ( ( y  e. Word  B  /\  <" z ">  e. Word  B )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
4933, 35, 48syl2anc 644 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
50 revs1 11799 . . . . . . . . . . 11  |-  (reverse `  <" z "> )  =  <" z ">
5150oveq1i 6093 . . . . . . . . . 10  |-  ( (reverse `  <" z "> ) concat  (reverse `  y
) )  =  (
<" z "> concat  (reverse `  y ) )
5249, 51syl6eq 2486 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( <" z "> concat  (reverse `  y )
) )
5352oveq2d 6099 . . . . . . . 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 445 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  M  e.  Mnd )
55 revcl 11795 . . . . . . . . . 10  |-  ( y  e. Word  B  ->  (reverse `  y )  e. Word  B
)
5655ad2antrl 710 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  y )  e. Word  B
)
5736, 44gsumccat 14789 . . . . . . . . 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 1185 . . . . . . . 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 14787 . . . . . . . . . 10  |-  ( z  e.  B  ->  ( M  gsumg 
<" z "> )  =  z )
6059ad2antll 711 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg 
<" z "> )  =  z )
6160oveq1d 6098 . . . . . . . 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 2474 . . . . . . 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 2452 . . . . . 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 214 . . . . 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 426 . . . 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 25 . . 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 11793 . 2  |-  ( W  e. Word  B  ->  ( M  e.  Mnd  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) ) )
6867impcom 421 1  |-  ( ( M  e.  Mnd  /\  W  e. Word  B )  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   (/)c0 3630   ` cfv 5456  (class class class)co 6083  Word cword 11719   concat cconcat 11720   <"cs1 11721  reversecreverse 11724   Basecbs 13471   +g cplusg 13531   0gc0g 13725    gsumg cgsu 13726   Mndcmnd 14686  oppgcoppg 15143
This theorem is referenced by:  symgtrinv  27392
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-tpos 6481  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-card 7828  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-fzo 11138  df-seq 11326  df-hash 11621  df-word 11725  df-concat 11726  df-s1 11727  df-substr 11728  df-reverse 11730  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-gsum 13730  df-mnd 14692  df-submnd 14741  df-oppg 15144
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