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Theorem oppgmnd 15991
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 2454 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2oppgbas 15988 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 11 . 2  |-  ( R  e.  Mnd  ->  ( Base `  R )  =  ( Base `  O
) )
5 eqidd 2455 . 2  |-  ( R  e.  Mnd  ->  ( +g  `  O )  =  ( +g  `  O
) )
6 eqid 2454 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
7 eqid 2454 . . . 4  |-  ( +g  `  O )  =  ( +g  `  O )
86, 1, 7oppgplus 15986 . . 3  |-  ( x ( +g  `  O
) y )  =  ( y ( +g  `  R ) x )
92, 6mndcl 15542 . . . 4  |-  ( ( R  e.  Mnd  /\  y  e.  ( Base `  R )  /\  x  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
1093com23 1194 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  (
y ( +g  `  R
) x )  e.  ( Base `  R
) )
118, 10syl5eqel 2546 . 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 996 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
z  e.  ( Base `  R ) )
14 simpr2 995 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  -> 
y  e.  ( Base `  R ) )
15 simpr1 994 . . . . 5  |-  ( ( R  e.  Mnd  /\  ( x  e.  ( Base `  R )  /\  y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
) ) )  ->  x  e.  ( Base `  R ) )
162, 6mndass 15543 . . . . 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 1221 . . . 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 2462 . . 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 6213 . . . 4  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( ( y ( +g  `  R ) x ) ( +g  `  O
) z )
206, 1, 7oppgplus 15986 . . . 4  |-  ( ( y ( +g  `  R
) x ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
2119, 20eqtri 2483 . . 3  |-  ( ( x ( +g  `  O
) y ) ( +g  `  O ) z )  =  ( z ( +g  `  R
) ( y ( +g  `  R ) x ) )
226, 1, 7oppgplus 15986 . . . . 5  |-  ( y ( +g  `  O
) z )  =  ( z ( +g  `  R ) y )
2322oveq2i 6214 . . . 4  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( x ( +g  `  O ) ( z ( +g  `  R
) y ) )
246, 1, 7oppgplus 15986 . . . 4  |-  ( x ( +g  `  O
) ( z ( +g  `  R ) y ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2523, 24eqtri 2483 . . 3  |-  ( x ( +g  `  O
) ( y ( +g  `  O ) z ) )  =  ( ( z ( +g  `  R ) y ) ( +g  `  R ) x )
2618, 21, 253eqtr4g 2520 . 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 2454 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
282, 27mndidcl 15561 . 2  |-  ( R  e.  Mnd  ->  ( 0g `  R )  e.  ( Base `  R
) )
296, 1, 7oppgplus 15986 . . 3  |-  ( ( 0g `  R ) ( +g  `  O
) x )  =  ( x ( +g  `  R ) ( 0g
`  R ) )
302, 6, 27mndrid 15564 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  R ) ( 0g
`  R ) )  =  x )
3129, 30syl5eq 2507 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  O ) x )  =  x )
326, 1, 7oppgplus 15986 . . 3  |-  ( x ( +g  `  O
) ( 0g `  R ) )  =  ( ( 0g `  R ) ( +g  `  R ) x )
332, 6, 27mndlid 15563 . . 3  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( ( 0g `  R ) ( +g  `  R ) x )  =  x )
3432, 33syl5eq 2507 . 2  |-  ( ( R  e.  Mnd  /\  x  e.  ( Base `  R ) )  -> 
( x ( +g  `  O ) ( 0g
`  R ) )  =  x )
354, 5, 11, 26, 28, 31, 34ismndd 15566 1  |-  ( R  e.  Mnd  ->  O  e.  Mnd )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   Basecbs 14295   +g cplusg 14360   0gc0g 14500   Mndcmnd 15531  oppgcoppg 15982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-tpos 6858  df-recs 6945  df-rdg 6979  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-0g 14502  df-mnd 15537  df-oppg 15983
This theorem is referenced by:  oppgmndb  15992  oppggrp  15994  gsumwrev  16003  gsumzoppg  16565  gsumzoppgOLD  16566  gsumzinv  16567  gsumzinvOLD  16568  oppgtmd  19803
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