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Theorem rngosubdi 33744
Description: Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringsubdi.1 |- G = ( 1st ` R )
ringsubdi.2 |- H = ( 2nd ` R )
ringsubdi.3 |- X = ran G
ringsubdi.4 |- D = ( /g ` G )
Assertion
Ref Expression
rngosubdi |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B D C ) ) = ( ( A H B ) D ( A H C ) ) )
Proof of Theorem rngosubdi
StepHypRef Expression
1 ringsubdi.1 . . . . 5 |- G = ( 1st ` R )
2 ringsubdi.3 . . . . 5 |- X = ran G
3 eqid 2622 . . . . 5 |- ( inv ` G ) = ( inv ` G )
4 ringsubdi.4 . . . . 5 |- D = ( /g ` G )
51, 2, 3, 4rngosub 33729 . . . 4 |- ( ( R e. RingOps /\ B e. X /\ C e. X ) -> ( B D C ) = ( B G ( ( inv ` G ) ` C ) ) )
653adant3r1 1274 . . 3 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( B D C ) = ( B G ( ( inv ` G ) ` C ) ) )
76oveq2d 6666 . 2 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B D C ) ) = ( A H ( B G ( ( inv ` G ) ` C ) ) ) )
8 ringsubdi.2 . . . . . . 7 |- H = ( 2nd ` R )
91, 8, 2rngocl 33700 . . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ B e. X ) -> ( A H B ) e. X )
1093adant3r3 1276 . . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H B ) e. X )
111, 8, 2rngocl 33700 . . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( A H C ) e. X )
12113adant3r2 1275 . . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H C ) e. X )
1310, 12jca 554 . . . 4 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H B ) e. X /\ ( A H C ) e. X ) )
141, 2, 3, 4rngosub 33729 . . . . 5 |- ( ( R e. RingOps /\ ( A H B ) e. X /\ ( A H C ) e. X ) -> ( ( A H B ) D ( A H C ) ) = ( ( A H B ) G ( ( inv ` G ) ` ( A H C ) ) ) )
15143expb 1266 . . . 4 |- ( ( R e. RingOps /\ ( ( A H B ) e. X /\ ( A H C ) e. X ) ) -> ( ( A H B ) D ( A H C ) ) = ( ( A H B ) G ( ( inv ` G ) ` ( A H C ) ) ) )
1613, 15syldan 487 . . 3 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H B ) D ( A H C ) ) = ( ( A H B ) G ( ( inv ` G ) ` ( A H C ) ) ) )
17 idd 24 . . . . . . 7 |- ( R e. RingOps -> ( A e. X -> A e. X ) )
18 idd 24 . . . . . . 7 |- ( R e. RingOps -> ( B e. X -> B e. X ) )
191, 2, 3rngonegcl 33726 . . . . . . . 8 |- ( ( R e. RingOps /\ C e. X ) -> ( ( inv ` G ) ` C ) e. X )
2019ex 450 . . . . . . 7 |- ( R e. RingOps -> ( C e. X -> ( ( inv ` G ) ` C ) e. X ) )
2117, 18, 203anim123d 1406 . . . . . 6 |- ( R e. RingOps -> ( ( A e. X /\ B e. X /\ C e. X ) -> ( A e. X /\ B e. X /\ ( ( inv ` G ) ` C ) e. X ) ) )
2221imp 445 . . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A e. X /\ B e. X /\ ( ( inv ` G ) ` C ) e. X ) )
231, 8, 2rngodi 33703 . . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ ( ( inv ` G ) ` C ) e. X ) ) -> ( A H ( B G ( ( inv ` G ) ` C ) ) ) = ( ( A H B ) G ( A H ( ( inv ` G ) ` C ) ) ) )
2422, 23syldan 487 . . . 4 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B G ( ( inv ` G ) ` C ) ) ) = ( ( A H B ) G ( A H ( ( inv ` G ) ` C ) ) ) )
251, 8, 2, 3rngonegrmul 33743 . . . . . 6 |- ( ( R e. RingOps /\ A e. X /\ C e. X ) -> ( ( inv ` G ) ` ( A H C ) ) = ( A H ( ( inv ` G ) ` C ) ) )
26253adant3r2 1275 . . . . 5 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( inv ` G ) ` ( A H C ) ) = ( A H ( ( inv ` G ) ` C ) ) )
2726oveq2d 6666 . . . 4 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H B ) G ( ( inv ` G ) ` ( A H C ) ) ) = ( ( A H B ) G ( A H ( ( inv ` G ) ` C ) ) ) )
2824, 27eqtr4d 2659 . . 3 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B G ( ( inv ` G ) ` C ) ) ) = ( ( A H B ) G ( ( inv ` G ) ` ( A H C ) ) ) )
2916, 28eqtr4d 2659 . 2 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( ( A H B ) D ( A H C ) ) = ( A H ( B G ( ( inv ` G ) ` C ) ) ) )
307, 29eqtr4d 2659 1 |- ( ( R e. RingOps /\ ( A e. X /\ B e. X /\ C e. X ) ) -> ( A H ( B D C ) ) = ( ( A H B ) D ( A H C ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ran crn 5115   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   invcgn 27345   /g cgs 27346   RingOpscrngo 33693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-grpo 27347  df-gid 27348  df-ginv 27349  df-gdiv 27350  df-ablo 27399  df-ass 33642  df-exid 33644  df-mgmOLD 33648  df-sgrOLD 33660  df-mndo 33666  df-rngo 33694
This theorem is referenced by:  dmncan1  33875
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