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Theorem rngosubdi 33362
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
rngosubdi ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Proof of Theorem rngosubdi
StepHypRef Expression
1 ringsubdi.1 . . . . 5 𝐺 = (1st𝑅)
2 ringsubdi.3 . . . . 5 𝑋 = ran 𝐺
3 eqid 2626 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔𝐺)
51, 2, 3, 4rngosub 33347 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
653adant3r1 1271 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
76oveq2d 6621 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))))
8 ringsubdi.2 . . . . . . 7 𝐻 = (2nd𝑅)
91, 8, 2rngocl 33318 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
1093adant3r3 1273 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
111, 8, 2rngocl 33318 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
12113adant3r2 1272 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
1310, 12jca 554 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋))
141, 2, 3, 4rngosub 33347 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
15143expb 1263 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
1613, 15syldan 487 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
17 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐴𝑋𝐴𝑋))
18 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐵𝑋𝐵𝑋))
191, 2, 3rngonegcl 33344 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐶𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
2019ex 450 . . . . . . 7 (𝑅 ∈ RingOps → (𝐶𝑋 → ((inv‘𝐺)‘𝐶) ∈ 𝑋))
2117, 18, 203anim123d 1403 . . . . . 6 (𝑅 ∈ RingOps → ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)))
2221imp 445 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
231, 8, 2rngodi 33321 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
2422, 23syldan 487 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
251, 8, 2, 3rngonegrmul 33361 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → ((inv‘𝐺)‘(𝐴𝐻𝐶)) = (𝐴𝐻((inv‘𝐺)‘𝐶)))
26253adant3r2 1272 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘(𝐴𝐻𝐶)) = (𝐴𝐻((inv‘𝐺)‘𝐶)))
2726oveq2d 6621 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
2824, 27eqtr4d 2663 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
2916, 28eqtr4d 2663 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))))
307, 29eqtr4d 2663 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1992  ran crn 5080  cfv 5850  (class class class)co 6605  1st c1st 7114  2nd c2nd 7115  invcgn 27185   /𝑔 cgs 27186  RingOpscrngo 33311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-grpo 27187  df-gid 27188  df-ginv 27189  df-gdiv 27190  df-ablo 27239  df-ass 33260  df-exid 33262  df-mgmOLD 33266  df-sgrOLD 33278  df-mndo 33284  df-rngo 33312
This theorem is referenced by:  dmncan1  33493
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