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Theorem opprring 13311
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
opprring  |-  ( R  e.  Ring  ->  O  e. 
Ring )

Proof of Theorem opprring
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . 3  |-  O  =  (oppr
`  R )
2 eqid 2187 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbasg 13308 . 2  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  O )
)
4 eqid 2187 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
51, 4oppraddg 13309 . 2  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  O ) )
6 eqidd 2188 . 2  |-  ( R  e.  Ring  ->  ( .r
`  O )  =  ( .r `  O
) )
7 ringgrp 13238 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
8 eqidd 2188 . . . 4  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  R )
)
95oveqdr 5916 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  O ) y ) )
108, 3, 9grppropd 12912 . . 3  |-  ( R  e.  Ring  ->  ( R  e.  Grp  <->  O  e.  Grp ) )
117, 10mpbid 147 . 2  |-  ( R  e.  Ring  ->  O  e. 
Grp )
12 eqid 2187 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2187 . . . 4  |-  ( .r
`  O )  =  ( .r `  O
)
142, 12, 1, 13opprmulg 13304 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  =  ( y ( .r
`  R ) x ) )
152, 12ringcl 13250 . . . 4  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
16153com23 1210 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
1714, 16eqeltrd 2264 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  e.  ( Base `  R
) )
18 simpl 109 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  R  e.  Ring )
19 simpr3 1006 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  z  e.  ( Base `  R
) )
20 simpr2 1005 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  y  e.  ( Base `  R
) )
21 simpr1 1004 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  x  e.  ( Base `  R
) )
222, 12ringass 13253 . . . 4  |-  ( ( 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 1250 . . 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 ) ) )
242, 12, 1, 13opprmulg 13304 . . . . . 6  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( y
( .r `  O
) z )  =  ( z ( .r
`  R ) y ) )
25243adant3r1 1213 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( .r `  O ) z )  =  ( z ( .r `  R ) y ) )
2625oveq2d 5904 . . . 4  |-  ( ( 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 ) ( z ( .r
`  R ) y ) ) )
272, 12ringcl 13250 . . . . . 6  |-  ( ( R  e.  Ring  /\  z  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( z
( .r `  R
) y )  e.  ( Base `  R
) )
2818, 19, 20, 27syl3anc 1248 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
z ( .r `  R ) y )  e.  ( Base `  R
) )
292, 12, 1, 13opprmulg 13304 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  ( z
( .r `  R
) y )  e.  ( Base `  R
) )  ->  (
x ( .r `  O ) ( z ( .r `  R
) y ) )  =  ( ( z ( .r `  R
) y ) ( .r `  R ) x ) )
3018, 21, 28, 29syl3anc 1248 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) ( z ( .r `  R
) y ) )  =  ( ( z ( .r `  R
) y ) ( .r `  R ) x ) )
3126, 30eqtrd 2220 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) ( y ( .r `  O
) z ) )  =  ( ( z ( .r `  R
) y ) ( .r `  R ) x ) )
3214oveq1d 5903 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( (
x ( .r `  O ) y ) ( .r `  O
) z )  =  ( ( y ( .r `  R ) x ) ( .r
`  O ) z ) )
33323adant3r3 1215 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  O ) y ) ( .r `  O ) z )  =  ( ( y ( .r `  R
) x ) ( .r `  O ) z ) )
34163adant3r3 1215 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( .r `  R ) x )  e.  ( Base `  R
) )
352, 12, 1, 13opprmulg 13304 . . . . 5  |-  ( ( R  e.  Ring  /\  (
y ( .r `  R ) x )  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( (
y ( .r `  R ) x ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( y ( .r `  R ) x ) ) )
3618, 34, 19, 35syl3anc 1248 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( y ( .r
`  R ) x ) ( .r `  O ) z )  =  ( z ( .r `  R ) ( y ( .r
`  R ) x ) ) )
3733, 36eqtrd 2220 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  O ) y ) ( .r `  O ) z )  =  ( z ( .r `  R ) ( y ( .r
`  R ) x ) ) )
3823, 31, 373eqtr4rd 2231 . 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 ) ) )
392, 4, 12ringdir 13256 . . . 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 ) ) )
4018, 20, 19, 21, 39syl13anc 1250 . . 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 ) ) )
412, 4ringacl 13267 . . . . 5  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( y
( +g  `  R ) z )  e.  (
Base `  R )
)
42413adant3r1 1213 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( +g  `  R
) z )  e.  ( Base `  R
) )
432, 12, 1, 13opprmulg 13304 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  ( y
( +g  `  R ) z )  e.  (
Base `  R )
)  ->  ( x
( .r `  O
) ( y ( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z ) ( .r
`  R ) x ) )
4418, 21, 42, 43syl3anc 1248 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) ( y ( +g  `  R
) z ) )  =  ( ( y ( +g  `  R
) z ) ( .r `  R ) x ) )
45143adant3r3 1215 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) y )  =  ( y ( .r `  R ) x ) )
462, 12, 1, 13opprmulg 13304 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) z )  =  ( z ( .r
`  R ) x ) )
47463adant3r2 1214 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) z )  =  ( z ( .r `  R ) x ) )
4845, 47oveq12d 5906 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  O ) y ) ( +g  `  R
) ( x ( .r `  O ) z ) )  =  ( ( y ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) x ) ) )
4940, 44, 483eqtr4d 2230 . 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 ) ) )
502, 4, 12ringdi 13255 . . . 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 ) ) )
5118, 19, 21, 20, 50syl13anc 1250 . . 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 ) ) )
522, 4ringacl 13267 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
53523adant3r3 1215 . . . 4  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
542, 12, 1, 13opprmulg 13304 . . . 4  |-  ( ( R  e.  Ring  /\  (
x ( +g  `  R
) y )  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( (
x ( +g  `  R
) y ) ( .r `  O ) z )  =  ( z ( .r `  R ) ( x ( +g  `  R
) y ) ) )
5518, 53, 19, 54syl3anc 1248 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( +g  `  R ) y ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( x ( +g  `  R
) y ) ) )
5647, 25oveq12d 5906 . . 3  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
( x ( .r
`  O ) z ) ( +g  `  R
) ( y ( .r `  O ) z ) )  =  ( ( z ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) y ) ) )
5751, 55, 563eqtr4d 2230 . 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 ) ) )
58 eqid 2187 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
592, 58ringidcl 13257 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
60 simpl 109 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  R  e.  Ring )
6160, 59syl 14 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  ( 1r `  R )  e.  ( Base `  R
) )
62 simpr 110 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  x  e.  ( Base `  R
) )
632, 12, 1, 13opprmulg 13304 . . . 4  |-  ( ( R  e.  Ring  /\  ( 1r `  R )  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  O ) x )  =  ( x ( .r `  R ) ( 1r
`  R ) ) )
6460, 61, 62, 63syl3anc 1248 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  O ) x )  =  ( x ( .r `  R ) ( 1r `  R
) ) )
652, 12, 58ringridm 13261 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  R ) ( 1r
`  R ) )  =  x )
6664, 65eqtrd 2220 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  O ) x )  =  x )
672, 12, 1, 13opprmulg 13304 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  ( 1r `  R )  e.  (
Base `  R )
)  ->  ( x
( .r `  O
) ( 1r `  R ) )  =  ( ( 1r `  R ) ( .r
`  R ) x ) )
6860, 62, 61, 67syl3anc 1248 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  O ) ( 1r
`  R ) )  =  ( ( 1r
`  R ) ( .r `  R ) x ) )
692, 12, 58ringlidm 13260 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
7068, 69eqtrd 2220 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  O ) ( 1r
`  R ) )  =  x )
713, 5, 6, 11, 17, 38, 49, 57, 59, 66, 70isringd 13278 1  |-  ( R  e.  Ring  ->  O  e. 
Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 979    = wceq 1363    e. wcel 2158   ` cfv 5228  (class class class)co 5888   Basecbs 12475   +g cplusg 12550   .rcmulr 12551   Grpcgrp 12896   1rcur 13196   Ringcrg 13233  opprcoppr 13300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-pre-ltirr 7936  ax-pre-lttrn 7938  ax-pre-ltadd 7940
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-tpos 6259  df-pnf 8007  df-mnf 8008  df-ltxr 8010  df-inn 8933  df-2 8991  df-3 8992  df-ndx 12478  df-slot 12479  df-base 12481  df-sets 12482  df-plusg 12563  df-mulr 12564  df-0g 12724  df-mgm 12793  df-sgrp 12826  df-mnd 12837  df-grp 12899  df-mgp 13163  df-ur 13197  df-ring 13235  df-oppr 13301
This theorem is referenced by:  opprringbg  13312  mulgass3  13317  1unit  13339  opprunitd  13342  crngunit  13343  unitmulcl  13345  unitgrp  13348  unitnegcl  13362  unitpropdg  13380  subrguss  13420  subrgunit  13423  isridl  13652  ridl0  13658  ridl1  13659
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