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Theorem opprrng 15413
Description: An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
Hypothesis
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprrng  |-  ( R  e.  Ring  ->  O  e. 
Ring )

Proof of Theorem opprrng
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . . 4  |-  O  =  (oppr
`  R )
2 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 15411 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 10 . 2  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  O )
)
5 eqid 2283 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
61, 5oppradd 15412 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
76a1i 10 . 2  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  O ) )
8 eqidd 2284 . 2  |-  ( R  e.  Ring  ->  ( .r
`  O )  =  ( .r `  O
) )
9 rnggrp 15346 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
103, 6grpprop 14501 . . 3  |-  ( R  e.  Grp  <->  O  e.  Grp )
119, 10sylib 188 . 2  |-  ( R  e.  Ring  ->  O  e. 
Grp )
12 eqid 2283 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2283 . . . 4  |-  ( .r
`  O )  =  ( .r `  O
)
142, 12, 1, 13opprmul 15408 . . 3  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
152, 12rngcl 15354 . . . 4  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
16153com23 1157 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
1714, 16syl5eqel 2367 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  e.  ( Base `  R
) )
18 simpl 443 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  R  e.  Ring )
19 simpr3 963 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  z  e.  ( Base `  R
) )
20 simpr2 962 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  y  e.  ( Base `  R
) )
21 simpr1 961 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  x  e.  ( Base `  R
) )
222, 12rngass 15357 . . . . 5  |-  ( ( R  e.  Ring  /\  (
z  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
) )  ->  (
( z ( .r
`  R ) y ) ( .r `  R ) x )  =  ( z ( .r `  R ) ( y ( .r
`  R ) x ) ) )
2318, 19, 20, 21, 22syl13anc 1184 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( z ( .r
`  R ) y ) ( .r `  R ) x )  =  ( z ( .r `  R ) ( y ( .r
`  R ) x ) ) )
2423eqcomd 2288 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
z ( .r `  R ) ( y ( .r `  R
) x ) )  =  ( ( z ( .r `  R
) y ) ( .r `  R ) x ) )
2514oveq1i 5868 . . . 4  |-  ( ( x ( .r `  O ) y ) ( .r `  O
) z )  =  ( ( y ( .r `  R ) x ) ( .r
`  O ) z )
262, 12, 1, 13opprmul 15408 . . . 4  |-  ( ( y ( .r `  R ) x ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( y ( .r `  R ) x ) )
2725, 26eqtri 2303 . . 3  |-  ( ( x ( .r `  O ) y ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( y ( .r `  R ) x ) )
282, 12, 1, 13opprmul 15408 . . . . 5  |-  ( y ( .r `  O
) z )  =  ( z ( .r
`  R ) y )
2928oveq2i 5869 . . . 4  |-  ( x ( .r `  O
) ( y ( .r `  O ) z ) )  =  ( x ( .r
`  O ) ( z ( .r `  R ) y ) )
302, 12, 1, 13opprmul 15408 . . . 4  |-  ( x ( .r `  O
) ( z ( .r `  R ) y ) )  =  ( ( z ( .r `  R ) y ) ( .r
`  R ) x )
3129, 30eqtri 2303 . . 3  |-  ( x ( .r `  O
) ( y ( .r `  O ) z ) )  =  ( ( z ( .r `  R ) y ) ( .r
`  R ) x )
3224, 27, 313eqtr4g 2340 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  O ) y ) ( .r `  O ) z )  =  ( x ( .r `  O ) ( y ( .r
`  O ) z ) ) )
332, 5, 12rngdir 15360 . . . 4  |-  ( ( R  e.  Ring  /\  (
y  e.  ( Base `  R )  /\  z  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
) )  ->  (
( y ( +g  `  R ) z ) ( .r `  R
) x )  =  ( ( y ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) x ) ) )
3418, 20, 19, 21, 33syl13anc 1184 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( y ( +g  `  R ) z ) ( .r `  R
) x )  =  ( ( y ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) x ) ) )
352, 12, 1, 13opprmul 15408 . . 3  |-  ( x ( .r `  O
) ( y ( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z ) ( .r
`  R ) x )
362, 12, 1, 13opprmul 15408 . . . 4  |-  ( x ( .r `  O
) z )  =  ( z ( .r
`  R ) x )
3714, 36oveq12i 5870 . . 3  |-  ( ( x ( .r `  O ) y ) ( +g  `  R
) ( x ( .r `  O ) z ) )  =  ( ( y ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) x ) )
3834, 35, 373eqtr4g 2340 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) ( y ( +g  `  R
) z ) )  =  ( ( x ( .r `  O
) y ) ( +g  `  R ) ( x ( .r
`  O ) z ) ) )
392, 5, 12rngdi 15359 . . . 4  |-  ( ( R  e.  Ring  /\  (
z  e.  ( Base `  R )  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
) )  ->  (
z ( .r `  R ) ( x ( +g  `  R
) y ) )  =  ( ( z ( .r `  R
) x ) ( +g  `  R ) ( z ( .r
`  R ) y ) ) )
4018, 19, 21, 20, 39syl13anc 1184 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
z ( .r `  R ) ( x ( +g  `  R
) y ) )  =  ( ( z ( .r `  R
) x ) ( +g  `  R ) ( z ( .r
`  R ) y ) ) )
412, 12, 1, 13opprmul 15408 . . 3  |-  ( ( x ( +g  `  R
) y ) ( .r `  O ) z )  =  ( z ( .r `  R ) ( x ( +g  `  R
) y ) )
4236, 28oveq12i 5870 . . 3  |-  ( ( x ( .r `  O ) z ) ( +g  `  R
) ( y ( .r `  O ) z ) )  =  ( ( z ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) y ) )
4340, 41, 423eqtr4g 2340 . 2  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( +g  `  R ) y ) ( .r `  O
) z )  =  ( ( x ( .r `  O ) z ) ( +g  `  R ) ( y ( .r `  O
) z ) ) )
44 eqid 2283 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
452, 44rngidcl 15361 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
462, 12, 1, 13opprmul 15408 . . 3  |-  ( ( 1r `  R ) ( .r `  O
) x )  =  ( x ( .r
`  R ) ( 1r `  R ) )
472, 12, 44rngridm 15365 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  R ) ( 1r
`  R ) )  =  x )
4846, 47syl5eq 2327 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  O ) x )  =  x )
492, 12, 1, 13opprmul 15408 . . 3  |-  ( x ( .r `  O
) ( 1r `  R ) )  =  ( ( 1r `  R ) ( .r
`  R ) x )
502, 12, 44rnglidm 15364 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
5149, 50syl5eq 2327 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  O ) ( 1r
`  R ) )  =  x )
524, 7, 8, 11, 17, 32, 38, 43, 45, 48, 51isrngd 15375 1  |-  ( R  e.  Ring  ->  O  e. 
Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   Grpcgrp 14362   Ringcrg 15337   1rcur 15339  opprcoppr 15404
This theorem is referenced by:  opprrngb  15414  mulgass3  15419  1unit  15440  unitmulcl  15446  unitnegcl  15463  irredlmul  15490  isdrngrd  15538  issrngd  15626  2idlcpbl  15986  opprnzr  16016  ply1divalg2  19524  lduallmodlem  29342
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-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-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-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405
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