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Theorem gsumwrev 14839
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 5866 . . . . 5  |-  ( x  =  (/)  ->  ( O 
gsumg  x )  =  ( O  gsumg  (/) ) )
2 fveq2 5525 . . . . . . 7  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (reverse `  (/) ) )
3 rev0 11482 . . . . . . 7  |-  (reverse `  (/) )  =  (/)
42, 3syl6eq 2331 . . . . . 6  |-  ( x  =  (/)  ->  (reverse `  x
)  =  (/) )
54oveq2d 5874 . . . . 5  |-  ( x  =  (/)  ->  ( M 
gsumg  (reverse `  x ) )  =  ( M  gsumg  (/) ) )
61, 5eqeq12d 2297 . . . 4  |-  ( x  =  (/)  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) ) )
76imbi2d 307 . . 3  |-  ( x  =  (/)  ->  ( ( M  e.  Mnd  ->  ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
) )  <->  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M 
gsumg  (/) ) ) ) )
8 oveq2 5866 . . . . 5  |-  ( x  =  y  ->  ( O  gsumg  x )  =  ( O  gsumg  y ) )
9 fveq2 5525 . . . . . 6  |-  ( x  =  y  ->  (reverse `  x )  =  (reverse `  y ) )
109oveq2d 5874 . . . . 5  |-  ( x  =  y  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  y ) ) )
118, 10eqeq12d 2297 . . . 4  |-  ( x  =  y  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y
) ) ) )
1211imbi2d 307 . . 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 5866 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( O  gsumg  x )  =  ( O  gsumg  ( y concat  <" z "> ) ) )
14 fveq2 5525 . . . . . 6  |-  ( x  =  ( y concat  <" z "> )  ->  (reverse `  x )  =  (reverse `  ( y concat  <" z "> ) ) )
1514oveq2d 5874 . . . . 5  |-  ( x  =  ( y concat  <" z "> )  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  ( y concat  <" z "> )
) ) )
1613, 15eqeq12d 2297 . . . 4  |-  ( x  =  ( y concat  <" z "> )  ->  ( ( O  gsumg  x )  =  ( M  gsumg  (reverse `  x
) )  <->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( M  gsumg  (reverse `  ( y concat  <" z "> ) ) ) ) )
1716imbi2d 307 . . 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 5866 . . . . 5  |-  ( x  =  W  ->  ( O  gsumg  x )  =  ( O  gsumg  W ) )
19 fveq2 5525 . . . . . 6  |-  ( x  =  W  ->  (reverse `  x )  =  (reverse `  W ) )
2019oveq2d 5874 . . . . 5  |-  ( x  =  W  ->  ( M  gsumg  (reverse `  x )
)  =  ( M 
gsumg  (reverse `  W ) ) )
2118, 20eqeq12d 2297 . . . 4  |-  ( x  =  W  ->  (
( O  gsumg  x )  =  ( M  gsumg  (reverse `  x )
)  <->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W
) ) ) )
2221imbi2d 307 . . 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 2283 . . . . . . 7  |-  ( 0g
`  M )  =  ( 0g `  M
)
2523, 24oppgid 14829 . . . . . 6  |-  ( 0g
`  M )  =  ( 0g `  O
)
2625gsum0 14457 . . . . 5  |-  ( O 
gsumg  (/) )  =  ( 0g
`  M )
2724gsum0 14457 . . . . 5  |-  ( M 
gsumg  (/) )  =  ( 0g
`  M )
2826, 27eqtr4i 2306 . . . 4  |-  ( O 
gsumg  (/) )  =  ( M 
gsumg  (/) )
2928a1i 10 . . 3  |-  ( M  e.  Mnd  ->  ( O  gsumg  (/) )  =  ( M  gsumg  (/) ) )
30 oveq2 5866 . . . . . 6  |-  ( ( O  gsumg  y )  =  ( M  gsumg  (reverse `  y )
)  ->  ( z
( +g  `  M ) ( O  gsumg  y ) )  =  ( z ( +g  `  M ) ( M 
gsumg  (reverse `  y ) ) ) )
3123oppgmnd 14827 . . . . . . . . . 10  |-  ( M  e.  Mnd  ->  O  e.  Mnd )
3231adantr 451 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  O  e.  Mnd )
33 simprl 732 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  y  e. Word  B )
34 simprr 733 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  z  e.  B )
3534s1cld 11442 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  <" z ">  e. Word  B )
36 gsumwrev.b . . . . . . . . . . 11  |-  B  =  ( Base `  M
)
3723, 36oppgbas 14824 . . . . . . . . . 10  |-  B  =  ( Base `  O
)
38 eqid 2283 . . . . . . . . . 10  |-  ( +g  `  O )  =  ( +g  `  O )
3937, 38gsumccat 14464 . . . . . . . . 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 1182 . . . . . . . 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 14462 . . . . . . . . . . 11  |-  ( z  e.  B  ->  ( O  gsumg 
<" z "> )  =  z )
4241ad2antll 709 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg 
<" z "> )  =  z )
4342oveq2d 5874 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
4544, 23, 38oppgplus 14822 . . . . . . . . 9  |-  ( ( O  gsumg  y ) ( +g  `  O ) z )  =  ( z ( +g  `  M ) ( O  gsumg  y ) )
4643, 45syl6eq 2331 . . . . . . . 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 2315 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( O  gsumg  ( y concat  <" z "> ) )  =  ( z ( +g  `  M ) ( O 
gsumg  y ) ) )
48 revccat 11484 . . . . . . . . . . 11  |-  ( ( y  e. Word  B  /\  <" z ">  e. Word  B )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
4933, 35, 48syl2anc 642 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( (reverse `  <" z "> ) concat  (reverse `  y ) ) )
50 revs1 11483 . . . . . . . . . . 11  |-  (reverse `  <" z "> )  =  <" z ">
5150oveq1i 5868 . . . . . . . . . 10  |-  ( (reverse `  <" z "> ) concat  (reverse `  y
) )  =  (
<" z "> concat  (reverse `  y ) )
5249, 51syl6eq 2331 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  ( y concat  <" z "> ) )  =  ( <" z "> concat  (reverse `  y )
) )
5352oveq2d 5874 . . . . . . . 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 443 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  M  e.  Mnd )
55 revcl 11479 . . . . . . . . . 10  |-  ( y  e. Word  B  ->  (reverse `  y )  e. Word  B
)
5655ad2antrl 708 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  (reverse `  y )  e. Word  B
)
5736, 44gsumccat 14464 . . . . . . . . 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 1182 . . . . . . . 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 14462 . . . . . . . . . 10  |-  ( z  e.  B  ->  ( M  gsumg 
<" z "> )  =  z )
6059ad2antll 709 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( y  e. Word  B  /\  z  e.  B
) )  ->  ( M  gsumg 
<" z "> )  =  z )
6160oveq1d 5873 . . . . . . . 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 2319 . . . . . . 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 2297 . . . . . 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 212 . . . . 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 424 . . . 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 23 . . 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 11477 . 2  |-  ( W  e. Word  B  ->  ( M  e.  Mnd  ->  ( O  gsumg  W )  =  ( M  gsumg  (reverse `  W )
) ) )
6867impcom 419 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 358    = wceq 1623    e. wcel 1684   (/)c0 3455   ` cfv 5255  (class class class)co 5858  Word cword 11403   concat cconcat 11404   <"cs1 11405  reversecreverse 11408   Basecbs 13148   +g cplusg 13208   0gc0g 13400    gsumg cgsu 13401   Mndcmnd 14361  oppgcoppg 14818
This theorem is referenced by:  symgtrinv  27413
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-rmo 2551  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-uni 3828  df-int 3863  df-iun 3907  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-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-word 11409  df-concat 11410  df-s1 11411  df-substr 11412  df-reverse 11414  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-gsum 13405  df-mnd 14367  df-submnd 14416  df-oppg 14819
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