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Theorem rngosub 31613
Description: Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringnegcl.1  |-  G  =  ( 1st `  R
)
ringnegcl.2  |-  X  =  ran  G
ringnegcl.3  |-  N  =  ( inv `  G
)
ringsub.4  |-  D  =  (  /g  `  G
)
Assertion
Ref Expression
rngosub  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )

Proof of Theorem rngosub
StepHypRef Expression
1 ringnegcl.1 . . 3  |-  G  =  ( 1st `  R
)
21rngogrpo 25792 . 2  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
3 ringnegcl.2 . . 3  |-  X  =  ran  G
4 ringnegcl.3 . . 3  |-  N  =  ( inv `  G
)
5 ringsub.4 . . 3  |-  D  =  (  /g  `  G
)
63, 4, 5grpodivval 25645 . 2  |-  ( ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
72, 6syl3an1 1263 1  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1405    e. wcel 1842   ran crn 4823   ` cfv 5568  (class class class)co 6277   1stc1st 6781   GrpOpcgr 25588   invcgn 25590    /g cgs 25591   RingOpscrngo 25777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-gdiv 25596  df-ablo 25684  df-rngo 25778
This theorem is referenced by:  rngosubdi  31618  rngosubdir  31619  idlsubcl  31682
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