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Theorem rngsubdir 13047
Description: Ring multiplication distributes over subtraction. (subdir 8320 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
Hypotheses
Ref Expression
ringsubdi.b  |-  B  =  ( Base `  R
)
ringsubdi.t  |-  .x.  =  ( .r `  R )
ringsubdi.m  |-  .-  =  ( -g `  R )
ringsubdi.r  |-  ( ph  ->  R  e.  Ring )
ringsubdi.x  |-  ( ph  ->  X  e.  B )
ringsubdi.y  |-  ( ph  ->  Y  e.  B )
ringsubdi.z  |-  ( ph  ->  Z  e.  B )
Assertion
Ref Expression
rngsubdir  |-  ( ph  ->  ( ( X  .-  Y )  .x.  Z
)  =  ( ( X  .x.  Z ) 
.-  ( Y  .x.  Z ) ) )

Proof of Theorem rngsubdir
StepHypRef Expression
1 ringsubdi.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 ringsubdi.x . . . 4  |-  ( ph  ->  X  e.  B )
3 ringgrp 12997 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
41, 3syl 14 . . . . 5  |-  ( ph  ->  R  e.  Grp )
5 ringsubdi.y . . . . 5  |-  ( ph  ->  Y  e.  B )
6 ringsubdi.b . . . . . 6  |-  B  =  ( Base `  R
)
7 eqid 2177 . . . . . 6  |-  ( invg `  R )  =  ( invg `  R )
86, 7grpinvcl 12798 . . . . 5  |-  ( ( R  e.  Grp  /\  Y  e.  B )  ->  ( ( invg `  R ) `  Y
)  e.  B )
94, 5, 8syl2anc 411 . . . 4  |-  ( ph  ->  ( ( invg `  R ) `  Y
)  e.  B )
10 ringsubdi.z . . . 4  |-  ( ph  ->  Z  e.  B )
11 eqid 2177 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
12 ringsubdi.t . . . . 5  |-  .x.  =  ( .r `  R )
136, 11, 12ringdir 13015 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( ( invg `  R ) `  Y
)  e.  B  /\  Z  e.  B )
)  ->  ( ( X ( +g  `  R
) ( ( invg `  R ) `
 Y ) ) 
.x.  Z )  =  ( ( X  .x.  Z ) ( +g  `  R ) ( ( ( invg `  R ) `  Y
)  .x.  Z )
) )
141, 2, 9, 10, 13syl13anc 1240 . . 3  |-  ( ph  ->  ( ( X ( +g  `  R ) ( ( invg `  R ) `  Y
) )  .x.  Z
)  =  ( ( X  .x.  Z ) ( +g  `  R
) ( ( ( invg `  R
) `  Y )  .x.  Z ) ) )
156, 12, 7, 1, 5, 10ringmneg1 13043 . . . 4  |-  ( ph  ->  ( ( ( invg `  R ) `
 Y )  .x.  Z )  =  ( ( invg `  R ) `  ( Y  .x.  Z ) ) )
1615oveq2d 5884 . . 3  |-  ( ph  ->  ( ( X  .x.  Z ) ( +g  `  R ) ( ( ( invg `  R ) `  Y
)  .x.  Z )
)  =  ( ( X  .x.  Z ) ( +g  `  R
) ( ( invg `  R ) `
 ( Y  .x.  Z ) ) ) )
1714, 16eqtrd 2210 . 2  |-  ( ph  ->  ( ( X ( +g  `  R ) ( ( invg `  R ) `  Y
) )  .x.  Z
)  =  ( ( X  .x.  Z ) ( +g  `  R
) ( ( invg `  R ) `
 ( Y  .x.  Z ) ) ) )
18 ringsubdi.m . . . . 5  |-  .-  =  ( -g `  R )
196, 11, 7, 18grpsubval 12796 . . . 4  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y
)  =  ( X ( +g  `  R
) ( ( invg `  R ) `
 Y ) ) )
202, 5, 19syl2anc 411 . . 3  |-  ( ph  ->  ( X  .-  Y
)  =  ( X ( +g  `  R
) ( ( invg `  R ) `
 Y ) ) )
2120oveq1d 5883 . 2  |-  ( ph  ->  ( ( X  .-  Y )  .x.  Z
)  =  ( ( X ( +g  `  R
) ( ( invg `  R ) `
 Y ) ) 
.x.  Z ) )
226, 12ringcl 13009 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
231, 2, 10, 22syl3anc 1238 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
246, 12ringcl 13009 . . . 4  |-  ( ( R  e.  Ring  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  .x.  Z )  e.  B )
251, 5, 10, 24syl3anc 1238 . . 3  |-  ( ph  ->  ( Y  .x.  Z
)  e.  B )
266, 11, 7, 18grpsubval 12796 . . 3  |-  ( ( ( X  .x.  Z
)  e.  B  /\  ( Y  .x.  Z )  e.  B )  -> 
( ( X  .x.  Z )  .-  ( Y  .x.  Z ) )  =  ( ( X 
.x.  Z ) ( +g  `  R ) ( ( invg `  R ) `  ( Y  .x.  Z ) ) ) )
2723, 25, 26syl2anc 411 . 2  |-  ( ph  ->  ( ( X  .x.  Z )  .-  ( Y  .x.  Z ) )  =  ( ( X 
.x.  Z ) ( +g  `  R ) ( ( invg `  R ) `  ( Y  .x.  Z ) ) ) )
2817, 21, 273eqtr4d 2220 1  |-  ( ph  ->  ( ( X  .-  Y )  .x.  Z
)  =  ( ( X  .x.  Z ) 
.-  ( Y  .x.  Z ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   ` cfv 5211  (class class class)co 5868   Basecbs 12432   +g cplusg 12505   .rcmulr 12506   Grpcgrp 12754   invgcminusg 12755   -gcsg 12756   Ringcrg 12992
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-pnf 7971  df-mnf 7972  df-ltxr 7974  df-inn 8896  df-2 8954  df-3 8955  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758  df-sbg 12759  df-mgp 12945  df-ur 12956  df-ring 12994
This theorem is referenced by: (None)
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