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Theorem ringsubdi 17443
Description: Ring multiplication distributes over subtraction. (subdi 9986 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
ringsubdi  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X 
.x.  Y )  .-  ( X  .x.  Z ) ) )

Proof of Theorem ringsubdi
StepHypRef Expression
1 ringsubdi.r . . . 4  |-  ( ph  ->  R  e.  Ring )
2 ringsubdi.x . . . 4  |-  ( ph  ->  X  e.  B )
3 ringsubdi.y . . . 4  |-  ( ph  ->  Y  e.  B )
4 ringgrp 17401 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
51, 4syl 16 . . . . 5  |-  ( ph  ->  R  e.  Grp )
6 ringsubdi.z . . . . 5  |-  ( ph  ->  Z  e.  B )
7 ringsubdi.b . . . . . 6  |-  B  =  ( Base `  R
)
8 eqid 2454 . . . . . 6  |-  ( invg `  R )  =  ( invg `  R )
97, 8grpinvcl 16297 . . . . 5  |-  ( ( R  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  R ) `  Z
)  e.  B )
105, 6, 9syl2anc 659 . . . 4  |-  ( ph  ->  ( ( invg `  R ) `  Z
)  e.  B )
11 eqid 2454 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
12 ringsubdi.t . . . . 5  |-  .x.  =  ( .r `  R )
137, 11, 12ringdi 17415 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( invg `  R ) `  Z
)  e.  B ) )  ->  ( X  .x.  ( Y ( +g  `  R ) ( ( invg `  R
) `  Z )
) )  =  ( ( X  .x.  Y
) ( +g  `  R
) ( X  .x.  ( ( invg `  R ) `  Z
) ) ) )
141, 2, 3, 10, 13syl13anc 1228 . . 3  |-  ( ph  ->  ( X  .x.  ( Y ( +g  `  R
) ( ( invg `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( X  .x.  ( ( invg `  R ) `  Z
) ) ) )
157, 12, 8, 1, 2, 6ringmneg2 17441 . . . 4  |-  ( ph  ->  ( X  .x.  (
( invg `  R ) `  Z
) )  =  ( ( invg `  R ) `  ( X  .x.  Z ) ) )
1615oveq2d 6286 . . 3  |-  ( ph  ->  ( ( X  .x.  Y ) ( +g  `  R ) ( X 
.x.  ( ( invg `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( ( invg `  R ) `
 ( X  .x.  Z ) ) ) )
1714, 16eqtrd 2495 . 2  |-  ( ph  ->  ( X  .x.  ( Y ( +g  `  R
) ( ( invg `  R ) `
 Z ) ) )  =  ( ( X  .x.  Y ) ( +g  `  R
) ( ( invg `  R ) `
 ( X  .x.  Z ) ) ) )
18 ringsubdi.m . . . . 5  |-  .-  =  ( -g `  R )
197, 11, 8, 18grpsubval 16295 . . . 4  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  R
) ( ( invg `  R ) `
 Z ) ) )
203, 6, 19syl2anc 659 . . 3  |-  ( ph  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  R
) ( ( invg `  R ) `
 Z ) ) )
2120oveq2d 6286 . 2  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( X  .x.  ( Y ( +g  `  R
) ( ( invg `  R ) `
 Z ) ) ) )
227, 12ringcl 17410 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  e.  B )
231, 2, 3, 22syl3anc 1226 . . 3  |-  ( ph  ->  ( X  .x.  Y
)  e.  B )
247, 12ringcl 17410 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  .x.  Z )  e.  B )
251, 2, 6, 24syl3anc 1226 . . 3  |-  ( ph  ->  ( X  .x.  Z
)  e.  B )
267, 11, 8, 18grpsubval 16295 . . 3  |-  ( ( ( X  .x.  Y
)  e.  B  /\  ( X  .x.  Z )  e.  B )  -> 
( ( X  .x.  Y )  .-  ( X  .x.  Z ) )  =  ( ( X 
.x.  Y ) ( +g  `  R ) ( ( invg `  R ) `  ( X  .x.  Z ) ) ) )
2723, 25, 26syl2anc 659 . 2  |-  ( ph  ->  ( ( X  .x.  Y )  .-  ( X  .x.  Z ) )  =  ( ( X 
.x.  Y ) ( +g  `  R ) ( ( invg `  R ) `  ( X  .x.  Z ) ) ) )
2817, 21, 273eqtr4d 2505 1  |-  ( ph  ->  ( X  .x.  ( Y  .-  Z ) )  =  ( ( X 
.x.  Y )  .-  ( X  .x.  Z ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270   Basecbs 14719   +g cplusg 14787   .rcmulr 14788   Grpcgrp 16255   invgcminusg 16256   -gcsg 16257   Ringcrg 17396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-plusg 14800  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mgp 17340  df-ur 17352  df-ring 17398
This theorem is referenced by:  2idlcpbl  18080  mdetuni0  19293  chfacfpmmulgsum2  19536  nrgdsdi  21343  nrginvrcnlem  21368  ply1divmo  22705  ornglmulle  28033
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