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Theorem oppgmnd 16261
Description: The opposite of a monoid is a monoid. (Contributed by Stefan O'Rear, 26-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.)
Hypothesis
Ref Expression
oppgbas.1  |-  O  =  (oppg
`  R )
Assertion
Ref Expression
oppgmnd  |-  ( R  e.  Mnd  ->  O  e.  Mnd )

Proof of Theorem oppgmnd
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppgbas.1 . . . 4  |-  O  =  (oppg
`  R )
2 eqid 2467 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2oppgbas 16258 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 11 . 2  |-  ( R  e.  Mnd  ->  ( Base `  R )  =  ( Base `  O
) )
5 eqidd 2468 . 2  |-  ( R  e.  Mnd  ->  ( +g  `  O )  =  ( +g  `  O
) )
6 eqid 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
7 eqid 2467 . . . 4  |-  ( +g  `  O )  =  ( +g  `  O )
86, 1, 7oppgplus 16256 . . 3  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
92, 6mndcl 15802 . . . 4  |-  ( ( R  e.  Mnd  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
1093com23 1202 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
118, 10syl5eqel 2559 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
x ( +g  `  O
) y )  e.  ( Base `  R
) )
12 simpl 457 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  R  e.  Mnd )
13 simpr3 1004 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
z  e.  ( Base `  R ) )
14 simpr2 1003 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
y  e.  ( Base `  R ) )
15 simpr1 1002 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  x  e.  ( Base `  R ) )
162, 6mndass 15803 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( z  e.  (
Base `  R )  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R ) ) )  ->  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) ) )
1712, 13, 14, 15, 16syl13anc 1230 . . . 4  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( z ( +g  `  R ) y ) ( +g  `  R ) x )  =  ( z ( +g  `  R ) ( y ( +g  `  R ) x ) ) )
1817eqcomd 2475 . . 3  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( z ( +g  `  R ) ( y ( +g  `  R
) x ) )  =  ( ( z ( +g  `  R
) y ) ( +g  `  R ) x ) )
198oveq1i 6305 . . . 4  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( ( y ( +g  `  R ) x ) ( +g  `  O
) z )
206, 1, 7oppgplus 16256 . . . 4  |-  ( ( y ( +g  `  R
) x ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
2119, 20eqtri 2496 . . 3  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
226, 1, 7oppgplus 16256 . . . . 5  |-  ( y ( +g  `  O
) z )  =  ( z ( +g  `  R ) y )
2322oveq2i 6306 . . . 4  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( x ( +g  `  O ) ( z ( +g  `  R
) y ) )
246, 1, 7oppgplus 16256 . . . 4  |-  ( x ( +g  `  O
) ( z ( +g  `  R ) y ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2523, 24eqtri 2496 . . 3  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2618, 21, 253eqtr4g 2533 . 2  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
( ( x ( +g  `  O ) y ) ( +g  `  O ) z )  =  ( x ( +g  `  O ) ( y ( +g  `  O ) z ) ) )
27 eqid 2467 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
282, 27mndidcl 15811 . 2  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  ( Base `  R
) )
296, 1, 7oppgplus 16256 . . 3  |-  ( ( 0g `  R ) ( +g  `  O
) x )  =  ( x ( +g  `  R ) ( 0g
`  R ) )
302, 6, 27mndrid 15815 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  R ) ( 0g
`  R ) )  =  x )
3129, 30syl5eq 2520 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  O ) x )  =  x )
326, 1, 7oppgplus 16256 . . 3  |-  ( x ( +g  `  O
) ( 0g `  R ) )  =  ( ( 0g `  R ) ( +g  `  R ) x )
332, 6, 27mndlid 15814 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  R ) x )  =  x )
3432, 33syl5eq 2520 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  O ) ( 0g
`  R ) )  =  x )
354, 5, 11, 26, 28, 31, 34ismndd 15816 1  |-  ( R  e.  Mnd  ->  O  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   Mndcmnd 15793  oppgcoppg 16252
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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-oppg 16253
This theorem is referenced by:  oppgmndb  16262  oppggrp  16264  gsumwrev  16273  gsumzoppg  16840  gsumzoppgOLD  16841  gsumzinv  16842  gsumzinvOLD  16843  oppgtmd  20464
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