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Theorem oppgid 15152
Description: Zero in a monoid is a symmetric notion. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypotheses
Ref Expression
oppgbas.1  |-  O  =  (oppg
`  R )
oppgid.2  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
oppgid  |-  .0.  =  ( 0g `  O )

Proof of Theorem oppgid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ancom 438 . . . . . 6  |-  ( ( ( x ( +g  `  R ) y )  =  y  /\  (
y ( +g  `  R
) x )  =  y )  <->  ( (
y ( +g  `  R
) x )  =  y  /\  ( x ( +g  `  R
) y )  =  y ) )
2 eqid 2436 . . . . . . . . 9  |-  ( +g  `  R )  =  ( +g  `  R )
3 oppgbas.1 . . . . . . . . 9  |-  O  =  (oppg
`  R )
4 eqid 2436 . . . . . . . . 9  |-  ( +g  `  O )  =  ( +g  `  O )
52, 3, 4oppgplus 15145 . . . . . . . 8  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
65eqeq1i 2443 . . . . . . 7  |-  ( ( x ( +g  `  O
) y )  =  y  <->  ( y ( +g  `  R ) x )  =  y )
72, 3, 4oppgplus 15145 . . . . . . . 8  |-  ( y ( +g  `  O
) x )  =  ( x ( +g  `  R ) y )
87eqeq1i 2443 . . . . . . 7  |-  ( ( y ( +g  `  O
) x )  =  y  <->  ( x ( +g  `  R ) y )  =  y )
96, 8anbi12i 679 . . . . . 6  |-  ( ( ( x ( +g  `  O ) y )  =  y  /\  (
y ( +g  `  O
) x )  =  y )  <->  ( (
y ( +g  `  R
) x )  =  y  /\  ( x ( +g  `  R
) y )  =  y ) )
101, 9bitr4i 244 . . . . 5  |-  ( ( ( x ( +g  `  R ) y )  =  y  /\  (
y ( +g  `  R
) x )  =  y )  <->  ( (
x ( +g  `  O
) y )  =  y  /\  ( y ( +g  `  O
) x )  =  y ) )
1110ralbii 2729 . . . 4  |-  ( A. y  e.  ( Base `  R ) ( ( x ( +g  `  R
) y )  =  y  /\  ( y ( +g  `  R
) x )  =  y )  <->  A. y  e.  ( Base `  R
) ( ( x ( +g  `  O
) y )  =  y  /\  ( y ( +g  `  O
) x )  =  y ) )
1211anbi2i 676 . . 3  |-  ( ( x  e.  ( Base `  R )  /\  A. y  e.  ( Base `  R ) ( ( x ( +g  `  R
) y )  =  y  /\  ( y ( +g  `  R
) x )  =  y ) )  <->  ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( +g  `  O
) y )  =  y  /\  ( y ( +g  `  O
) x )  =  y ) ) )
1312iotabii 5440 . 2  |-  ( iota
x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( +g  `  R
) y )  =  y  /\  ( y ( +g  `  R
) x )  =  y ) ) )  =  ( iota x
( x  e.  (
Base `  R )  /\  A. y  e.  (
Base `  R )
( ( x ( +g  `  O ) y )  =  y  /\  ( y ( +g  `  O ) x )  =  y ) ) )
14 eqid 2436 . . 3  |-  ( Base `  R )  =  (
Base `  R )
15 oppgid.2 . . 3  |-  .0.  =  ( 0g `  R )
1614, 2, 15grpidval 14707 . 2  |-  .0.  =  ( iota x ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( +g  `  R
) y )  =  y  /\  ( y ( +g  `  R
) x )  =  y ) ) )
173, 14oppgbas 15147 . . 3  |-  ( Base `  R )  =  (
Base `  O )
18 eqid 2436 . . 3  |-  ( 0g
`  O )  =  ( 0g `  O
)
1917, 4, 18grpidval 14707 . 2  |-  ( 0g
`  O )  =  ( iota x ( x  e.  ( Base `  R )  /\  A. y  e.  ( Base `  R ) ( ( x ( +g  `  O
) y )  =  y  /\  ( y ( +g  `  O
) x )  =  y ) ) )
2013, 16, 193eqtr4i 2466 1  |-  .0.  =  ( 0g `  O )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   iotacio 5416   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723  oppgcoppg 15141
This theorem is referenced by:  oppggrp  15153  oppginv  15155  oppgsubm  15158  gsumwrev  15162  lsmdisj2r  15317  gsumzoppg  15539  tgpconcomp  18142
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-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-uni 4016  df-iun 4095  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-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  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-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-0g 13727  df-oppg 15142
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