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Theorem opprrng 16829
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 2451 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbas 16827 . . 3  |-  ( Base `  R )  =  (
Base `  O )
43a1i 11 . 2  |-  ( R  e.  Ring  ->  ( Base `  R )  =  (
Base `  O )
)
5 eqid 2451 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
61, 5oppradd 16828 . . 3  |-  ( +g  `  R )  =  ( +g  `  O )
76a1i 11 . 2  |-  ( R  e.  Ring  ->  ( +g  `  R )  =  ( +g  `  O ) )
8 eqidd 2452 . 2  |-  ( R  e.  Ring  ->  ( .r
`  O )  =  ( .r `  O
) )
9 rnggrp 16756 . . 3  |-  ( R  e.  Ring  ->  R  e. 
Grp )
103, 6grpprop 15659 . . 3  |-  ( R  e.  Grp  <->  O  e.  Grp )
119, 10sylib 196 . 2  |-  ( R  e.  Ring  ->  O  e. 
Grp )
12 eqid 2451 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2451 . . . 4  |-  ( .r
`  O )  =  ( .r `  O
)
142, 12, 1, 13opprmul 16824 . . 3  |-  ( x ( .r `  O
) y )  =  ( y ( .r
`  R ) x )
152, 12rngcl 16764 . . . 4  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
16153com23 1194 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
1714, 16syl5eqel 2543 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  e.  ( Base `  R
) )
18 simpl 457 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  R  e.  Ring )
19 simpr3 996 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  z  e.  ( Base `  R
) )
20 simpr2 995 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  y  e.  ( Base `  R
) )
21 simpr1 994 . . . . 5  |-  ( ( R  e.  Ring  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  x  e.  ( Base `  R
) )
222, 12rngass 16767 . . . . 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 1221 . . . 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 2459 . . 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 6200 . . . 4  |-  ( ( x ( .r `  O ) y ) ( .r `  O
) z )  =  ( ( y ( .r `  R ) x ) ( .r
`  O ) z )
262, 12, 1, 13opprmul 16824 . . . 4  |-  ( ( y ( .r `  R ) x ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( y ( .r `  R ) x ) )
2725, 26eqtri 2480 . . 3  |-  ( ( x ( .r `  O ) y ) ( .r `  O
) z )  =  ( z ( .r
`  R ) ( y ( .r `  R ) x ) )
282, 12, 1, 13opprmul 16824 . . . . 5  |-  ( y ( .r `  O
) z )  =  ( z ( .r
`  R ) y )
2928oveq2i 6201 . . . 4  |-  ( x ( .r `  O
) ( y ( .r `  O ) z ) )  =  ( x ( .r
`  O ) ( z ( .r `  R ) y ) )
302, 12, 1, 13opprmul 16824 . . . 4  |-  ( x ( .r `  O
) ( z ( .r `  R ) y ) )  =  ( ( z ( .r `  R ) y ) ( .r
`  R ) x )
3129, 30eqtri 2480 . . 3  |-  ( x ( .r `  O
) ( y ( .r `  O ) z ) )  =  ( ( z ( .r `  R ) y ) ( .r
`  R ) x )
3224, 27, 313eqtr4g 2517 . 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 16770 . . . 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 1221 . . 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 16824 . . 3  |-  ( x ( .r `  O
) ( y ( +g  `  R ) z ) )  =  ( ( y ( +g  `  R ) z ) ( .r
`  R ) x )
362, 12, 1, 13opprmul 16824 . . . 4  |-  ( x ( .r `  O
) z )  =  ( z ( .r
`  R ) x )
3714, 36oveq12i 6202 . . 3  |-  ( ( x ( .r `  O ) y ) ( +g  `  R
) ( x ( .r `  O ) z ) )  =  ( ( y ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) x ) )
3834, 35, 373eqtr4g 2517 . 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 16769 . . . 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 1221 . . 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 16824 . . 3  |-  ( ( x ( +g  `  R
) y ) ( .r `  O ) z )  =  ( z ( .r `  R ) ( x ( +g  `  R
) y ) )
4236, 28oveq12i 6202 . . 3  |-  ( ( x ( .r `  O ) z ) ( +g  `  R
) ( y ( .r `  O ) z ) )  =  ( ( z ( .r `  R ) x ) ( +g  `  R ) ( z ( .r `  R
) y ) )
4340, 41, 423eqtr4g 2517 . 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 2451 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
452, 44rngidcl 16771 . 2  |-  ( R  e.  Ring  ->  ( 1r
`  R )  e.  ( Base `  R
) )
462, 12, 1, 13opprmul 16824 . . 3  |-  ( ( 1r `  R ) ( .r `  O
) x )  =  ( x ( .r
`  R ) ( 1r `  R ) )
472, 12, 44rngridm 16775 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
x ( .r `  R ) ( 1r
`  R ) )  =  x )
4846, 47syl5eq 2504 . 2  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  O ) x )  =  x )
492, 12, 1, 13opprmul 16824 . . 3  |-  ( x ( .r `  O
) ( 1r `  R ) )  =  ( ( 1r `  R ) ( .r
`  R ) x )
502, 12, 44rnglidm 16774 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) x )  =  x )
5149, 50syl5eq 2504 . 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 16785 1  |-  ( R  e.  Ring  ->  O  e. 
Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5516  (class class class)co 6190   Basecbs 14276   +g cplusg 14340   .rcmulr 14341   Grpcgrp 15512   1rcur 16708   Ringcrg 16751  opprcoppr 16820
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 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-tpos 6845  df-recs 6932  df-rdg 6966  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-plusg 14353  df-mulr 14354  df-0g 14482  df-mnd 15517  df-grp 15647  df-mgp 16697  df-ur 16709  df-rng 16753  df-oppr 16821
This theorem is referenced by:  opprrngb  16830  mulgass3  16835  1unit  16856  unitmulcl  16862  unitnegcl  16879  irredlmul  16906  isdrngrd  16964  issrngd  17052  2idlcpbl  17422  opprnzr  17452  ply1divalg2  21726  lduallmodlem  33103
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