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Theorem opprrng 13424
Description: An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
Hypothesis
Ref Expression
opprbas.1  |-  O  =  (oppr
`  R )
Assertion
Ref Expression
opprrng  |-  ( R  e. Rng  ->  O  e. Rng )

Proof of Theorem opprrng
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 2189 . . 3  |-  ( Base `  R )  =  (
Base `  R )
31, 2opprbasg 13422 . 2  |-  ( R  e. Rng  ->  ( Base `  R
)  =  ( Base `  O ) )
4 eqid 2189 . . 3  |-  ( +g  `  R )  =  ( +g  `  R )
51, 4oppraddg 13423 . 2  |-  ( R  e. Rng  ->  ( +g  `  R
)  =  ( +g  `  O ) )
6 eqidd 2190 . 2  |-  ( R  e. Rng  ->  ( .r `  O )  =  ( .r `  O ) )
7 rngabl 13286 . . 3  |-  ( R  e. Rng  ->  R  e.  Abel )
8 eqidd 2190 . . . 4  |-  ( R  e. Rng  ->  ( Base `  R
)  =  ( Base `  R ) )
95oveqdr 5923 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) ) )  -> 
( x ( +g  `  R ) y )  =  ( x ( +g  `  O ) y ) )
108, 3, 9ablpropd 13232 . . 3  |-  ( R  e. Rng  ->  ( R  e. 
Abel 
<->  O  e.  Abel )
)
117, 10mpbid 147 . 2  |-  ( R  e. Rng  ->  O  e.  Abel )
12 eqid 2189 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2189 . . . 4  |-  ( .r
`  O )  =  ( .r `  O
)
142, 12, 1, 13opprmulg 13418 . . 3  |-  ( ( R  e. Rng  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  =  ( y ( .r
`  R ) x ) )
152, 12rngcl 13295 . . . 4  |-  ( ( R  e. Rng  /\  y  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
16153com23 1211 . . 3  |-  ( ( R  e. Rng  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( y
( .r `  R
) x )  e.  ( Base `  R
) )
1714, 16eqeltrd 2266 . 2  |-  ( ( R  e. Rng  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) y )  e.  ( Base `  R
) )
18 simpl 109 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  R  e. Rng )
19 simpr3 1007 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  z  e.  ( Base `  R
) )
20 simpr2 1006 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  y  e.  ( Base `  R
) )
21 simpr1 1005 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  x  e.  ( Base `  R
) )
222, 12rngass 13290 . . . 4  |-  ( ( R  e. Rng  /\  (
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 1251 . . 3  |-  ( ( R  e. Rng  /\  (
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 13418 . . . . . 6  |-  ( ( R  e. Rng  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( y
( .r `  O
) z )  =  ( z ( .r
`  R ) y ) )
25243adant3r1 1214 . . . . 5  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( .r `  O ) z )  =  ( z ( .r `  R ) y ) )
2625oveq2d 5911 . . . 4  |-  ( ( R  e. Rng  /\  (
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, 12rngcl 13295 . . . . . 6  |-  ( ( R  e. Rng  /\  z  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( z
( .r `  R
) y )  e.  ( Base `  R
) )
2818, 19, 20, 27syl3anc 1249 . . . . 5  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
z ( .r `  R ) y )  e.  ( Base `  R
) )
292, 12, 1, 13opprmulg 13418 . . . . 5  |-  ( ( R  e. Rng  /\  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 1249 . . . 4  |-  ( ( R  e. Rng  /\  (
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 2222 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) )
32143adant3r3 1216 . . . . 5  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) y )  =  ( y ( .r `  R ) x ) )
3332oveq1d 5910 . . . 4  |-  ( ( R  e. Rng  /\  (
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 ) )
3418, 20, 21, 15syl3anc 1249 . . . . 5  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( .r `  R ) x )  e.  ( Base `  R
) )
352, 12, 1, 13opprmulg 13418 . . . . 5  |-  ( ( R  e. Rng  /\  (
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 1249 . . . 4  |-  ( ( R  e. Rng  /\  (
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 2222 . . 3  |-  ( ( R  e. Rng  /\  (
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 2233 . 2  |-  ( ( R  e. Rng  /\  (
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, 12rngdir 13292 . . . 4  |-  ( ( R  e. Rng  /\  (
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 1251 . . 3  |-  ( ( R  e. Rng  /\  (
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, 4rngacl 13293 . . . . 5  |-  ( ( R  e. Rng  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( y
( +g  `  R ) z )  e.  (
Base `  R )
)
42413adant3r1 1214 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
y ( +g  `  R
) z )  e.  ( Base `  R
) )
432, 12, 1, 13opprmulg 13418 . . . 4  |-  ( ( R  e. Rng  /\  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 1249 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) )
452, 12, 1, 13opprmulg 13418 . . . . 5  |-  ( ( R  e. Rng  /\  x  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
)  ->  ( x
( .r `  O
) z )  =  ( z ( .r
`  R ) x ) )
4618, 21, 19, 45syl3anc 1249 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( .r `  O ) z )  =  ( z ( .r `  R ) x ) )
4732, 46oveq12d 5913 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) ) )
4840, 44, 473eqtr4d 2232 . 2  |-  ( ( R  e. Rng  /\  (
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 ) ) )
492, 4, 12rngdi 13291 . . . 4  |-  ( ( R  e. Rng  /\  (
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 ) ) )
5018, 19, 21, 20, 49syl13anc 1251 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) ) )
512, 4rngacl 13293 . . . . 5  |-  ( ( R  e. Rng  /\  x  e.  ( Base `  R
)  /\  y  e.  ( Base `  R )
)  ->  ( x
( +g  `  R ) y )  e.  (
Base `  R )
)
52513adant3r3 1216 . . . 4  |-  ( ( R  e. Rng  /\  (
x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
)  /\  z  e.  ( Base `  R )
) )  ->  (
x ( +g  `  R
) y )  e.  ( Base `  R
) )
532, 12, 1, 13opprmulg 13418 . . . 4  |-  ( ( R  e. Rng  /\  (
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 ) ) )
5418, 52, 19, 53syl3anc 1249 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) ) )
5546, 25oveq12d 5913 . . 3  |-  ( ( R  e. Rng  /\  (
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 ) ) )
5650, 54, 553eqtr4d 2232 . 2  |-  ( ( R  e. Rng  /\  (
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 ) ) )
573, 5, 6, 11, 17, 38, 48, 56isrngd 13304 1  |-  ( R  e. Rng  ->  O  e. Rng )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    = wceq 1364    e. wcel 2160   ` cfv 5235  (class class class)co 5895   Basecbs 12511   +g cplusg 12586   .rcmulr 12587   Abelcabl 13221  Rngcrng 13283  opprcoppr 13414
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-addcom 7940  ax-addass 7942  ax-i2m1 7945  ax-0lt1 7946  ax-0id 7948  ax-rnegex 7949  ax-pre-ltirr 7952  ax-pre-lttrn 7954  ax-pre-ltadd 7956
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-tpos 6269  df-pnf 8023  df-mnf 8024  df-ltxr 8026  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-plusg 12599  df-mulr 12600  df-0g 12760  df-mgm 12829  df-sgrp 12862  df-mnd 12875  df-grp 12945  df-cmn 13222  df-abl 13223  df-mgp 13272  df-rng 13284  df-oppr 13415
This theorem is referenced by:  opprrngbg  13425  opprsubrngg  13555  isridlrng  13795  2idlcpblrng  13835
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