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Theorem rngosubdir 34077
Description: Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
ringsubdi.1 𝐺 = (1st𝑅)
ringsubdi.2 𝐻 = (2nd𝑅)
ringsubdi.3 𝑋 = ran 𝐺
ringsubdi.4 𝐷 = ( /𝑔𝐺)
Assertion
Ref Expression
rngosubdir ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Proof of Theorem rngosubdir
StepHypRef Expression
1 ringsubdi.1 . . . . 5 𝐺 = (1st𝑅)
2 ringsubdi.3 . . . . 5 𝑋 = ran 𝐺
3 eqid 2771 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔𝐺)
51, 2, 3, 4rngosub 34061 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
653adant3r3 1199 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
76oveq1d 6811 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
8 ringsubdi.2 . . . . . . 7 𝐻 = (2nd𝑅)
91, 8, 2rngocl 34032 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
1093adant3r2 1198 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
111, 8, 2rngocl 34032 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) ∈ 𝑋)
12113adant3r1 1197 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) ∈ 𝑋)
1310, 12jca 501 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋))
141, 2, 3, 4rngosub 34061 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
15143expb 1113 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
1613, 15syldan 579 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
17 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐴𝑋𝐴𝑋))
181, 2, 3rngonegcl 34058 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
1918ex 397 . . . . . . 7 (𝑅 ∈ RingOps → (𝐵𝑋 → ((inv‘𝐺)‘𝐵) ∈ 𝑋))
20 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐶𝑋𝐶𝑋))
2117, 19, 203anim123d 1554 . . . . . 6 (𝑅 ∈ RingOps → ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋)))
2221imp 393 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋))
231, 8, 2rngodir 34036 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
2422, 23syldan 579 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
251, 8, 2, 3rngoneglmul 34074 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐵𝑋𝐶𝑋) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
26253adant3r1 1197 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
2726oveq2d 6812 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
2824, 27eqtr4d 2808 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
2916, 28eqtr4d 2808 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
307, 29eqtr4d 2808 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5251  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  invcgn 27685   /𝑔 cgs 27686  RingOpscrngo 34025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-grpo 27687  df-gid 27688  df-ginv 27689  df-gdiv 27690  df-ablo 27739  df-ass 33974  df-exid 33976  df-mgmOLD 33980  df-sgrOLD 33992  df-mndo 33998  df-rngo 34026
This theorem is referenced by: (None)
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