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Theorem rngmneg2 13295
Description: Negation of a product in a non-unital ring (mulneg2 8378 analog). In contrast to ringmneg2 13399, the proof does not (and cannot) make use of the existence of a ring unity. (Contributed by AV, 17-Feb-2025.)
Hypotheses
Ref Expression
rngneglmul.b  |-  B  =  ( Base `  R
)
rngneglmul.t  |-  .x.  =  ( .r `  R )
rngneglmul.n  |-  N  =  ( invg `  R )
rngneglmul.r  |-  ( ph  ->  R  e. Rng )
rngneglmul.x  |-  ( ph  ->  X  e.  B )
rngneglmul.y  |-  ( ph  ->  Y  e.  B )
Assertion
Ref Expression
rngmneg2  |-  ( ph  ->  ( X  .x.  ( N `  Y )
)  =  ( N `
 ( X  .x.  Y ) ) )

Proof of Theorem rngmneg2
StepHypRef Expression
1 rngneglmul.b . . . . . 6  |-  B  =  ( Base `  R
)
2 eqid 2189 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
3 eqid 2189 . . . . . 6  |-  ( 0g
`  R )  =  ( 0g `  R
)
4 rngneglmul.n . . . . . 6  |-  N  =  ( invg `  R )
5 rngneglmul.r . . . . . . 7  |-  ( ph  ->  R  e. Rng )
6 rnggrp 13285 . . . . . . 7  |-  ( R  e. Rng  ->  R  e.  Grp )
75, 6syl 14 . . . . . 6  |-  ( ph  ->  R  e.  Grp )
8 rngneglmul.y . . . . . 6  |-  ( ph  ->  Y  e.  B )
91, 2, 3, 4, 7, 8grplinvd 12992 . . . . 5  |-  ( ph  ->  ( ( N `  Y ) ( +g  `  R ) Y )  =  ( 0g `  R ) )
109oveq2d 5908 . . . 4  |-  ( ph  ->  ( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) )  =  ( X  .x.  ( 0g `  R ) ) )
11 rngneglmul.x . . . . 5  |-  ( ph  ->  X  e.  B )
12 rngneglmul.t . . . . . 6  |-  .x.  =  ( .r `  R )
131, 12, 3rngrz 13293 . . . . 5  |-  ( ( R  e. Rng  /\  X  e.  B )  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R
) )
145, 11, 13syl2anc 411 . . . 4  |-  ( ph  ->  ( X  .x.  ( 0g `  R ) )  =  ( 0g `  R ) )
1510, 14eqtrd 2222 . . 3  |-  ( ph  ->  ( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) )  =  ( 0g `  R ) )
161, 12rngcl 13291 . . . . . 6  |-  ( ( R  e. Rng  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
175, 11, 8, 16syl3anc 1249 . . . . 5  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
181, 4, 7, 8grpinvcld 12986 . . . . . 6  |-  ( ph  ->  ( N `  Y
)  e.  B )
191, 12rngcl 13291 . . . . . 6  |-  ( ( R  e. Rng  /\  X  e.  B  /\  ( N `  Y )  e.  B )  ->  ( X  .x.  ( N `  Y ) )  e.  B )
205, 11, 18, 19syl3anc 1249 . . . . 5  |-  ( ph  ->  ( X  .x.  ( N `  Y )
)  e.  B )
211, 2, 3, 4grpinvid2 12990 . . . . 5  |-  ( ( R  e.  Grp  /\  ( X  .x.  Y )  e.  B  /\  ( X  .x.  ( N `  Y ) )  e.  B )  ->  (
( N `  ( X  .x.  Y ) )  =  ( X  .x.  ( N `  Y ) )  <->  ( ( X 
.x.  ( N `  Y ) ) ( +g  `  R ) ( X  .x.  Y
) )  =  ( 0g `  R ) ) )
227, 17, 20, 21syl3anc 1249 . . . 4  |-  ( ph  ->  ( ( N `  ( X  .x.  Y ) )  =  ( X 
.x.  ( N `  Y ) )  <->  ( ( X  .x.  ( N `  Y ) ) ( +g  `  R ) ( X  .x.  Y
) )  =  ( 0g `  R ) ) )
231, 2, 12rngdi 13287 . . . . . . 7  |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  ( N `  Y )  e.  B  /\  Y  e.  B ) )  -> 
( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) )  =  ( ( X 
.x.  ( N `  Y ) ) ( +g  `  R ) ( X  .x.  Y
) ) )
2423eqcomd 2195 . . . . . 6  |-  ( ( R  e. Rng  /\  ( X  e.  B  /\  ( N `  Y )  e.  B  /\  Y  e.  B ) )  -> 
( ( X  .x.  ( N `  Y ) ) ( +g  `  R
) ( X  .x.  Y ) )  =  ( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) ) )
255, 11, 18, 8, 24syl13anc 1251 . . . . 5  |-  ( ph  ->  ( ( X  .x.  ( N `  Y ) ) ( +g  `  R
) ( X  .x.  Y ) )  =  ( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) ) )
2625eqeq1d 2198 . . . 4  |-  ( ph  ->  ( ( ( X 
.x.  ( N `  Y ) ) ( +g  `  R ) ( X  .x.  Y
) )  =  ( 0g `  R )  <-> 
( X  .x.  (
( N `  Y
) ( +g  `  R
) Y ) )  =  ( 0g `  R ) ) )
2722, 26bitrd 188 . . 3  |-  ( ph  ->  ( ( N `  ( X  .x.  Y ) )  =  ( X 
.x.  ( N `  Y ) )  <->  ( X  .x.  ( ( N `  Y ) ( +g  `  R ) Y ) )  =  ( 0g
`  R ) ) )
2815, 27mpbird 167 . 2  |-  ( ph  ->  ( N `  ( X  .x.  Y ) )  =  ( X  .x.  ( N `  Y ) ) )
2928eqcomd 2195 1  |-  ( ph  ->  ( X  .x.  ( N `  Y )
)  =  ( N `
 ( X  .x.  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 980    = wceq 1364    e. wcel 2160   ` cfv 5232  (class class class)co 5892   Basecbs 12507   +g cplusg 12582   .rcmulr 12583   0gc0g 12754   Grpcgrp 12938   invgcminusg 12939  Rngcrng 13279
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-coll 4133  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-addcom 7936  ax-addass 7938  ax-i2m1 7941  ax-0lt1 7942  ax-0id 7944  ax-rnegex 7945  ax-pre-ltirr 7948  ax-pre-ltadd 7952
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-reu 2475  df-rmo 2476  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-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-ltxr 8022  df-inn 8945  df-2 9003  df-3 9004  df-ndx 12510  df-slot 12511  df-base 12513  df-sets 12514  df-plusg 12595  df-mulr 12596  df-0g 12756  df-mgm 12825  df-sgrp 12858  df-mnd 12871  df-grp 12941  df-minusg 12942  df-abl 13219  df-mgp 13268  df-rng 13280
This theorem is referenced by:  rngm2neg  13296  rngsubdi  13298
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