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Theorem rngosubdir 35841
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 2737 . . . . 5 (inv‘𝐺) = (inv‘𝐺)
4 ringsubdi.4 . . . . 5 𝐷 = ( /𝑔𝐺)
51, 2, 3, 4rngosub 35825 . . . 4 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
653adant3r3 1186 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
76oveq1d 7228 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
8 ringsubdi.2 . . . . . . 7 𝐻 = (2nd𝑅)
91, 8, 2rngocl 35796 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
1093adant3r2 1185 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
111, 8, 2rngocl 35796 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) ∈ 𝑋)
12113adant3r1 1184 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) ∈ 𝑋)
1310, 12jca 515 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋))
141, 2, 3, 4rngosub 35825 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
15143expb 1122 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
1613, 15syldan 594 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
17 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐴𝑋𝐴𝑋))
181, 2, 3rngonegcl 35822 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐵𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
1918ex 416 . . . . . . 7 (𝑅 ∈ RingOps → (𝐵𝑋 → ((inv‘𝐺)‘𝐵) ∈ 𝑋))
20 idd 24 . . . . . . 7 (𝑅 ∈ RingOps → (𝐶𝑋𝐶𝑋))
2117, 19, 203anim123d 1445 . . . . . 6 (𝑅 ∈ RingOps → ((𝐴𝑋𝐵𝑋𝐶𝑋) → (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋)))
2221imp 410 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋))
231, 8, 2rngodir 35800 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
2422, 23syldan 594 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
251, 8, 2, 3rngoneglmul 35838 . . . . . 6 ((𝑅 ∈ RingOps ∧ 𝐵𝑋𝐶𝑋) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
26253adant3r1 1184 . . . . 5 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
2726oveq2d 7229 . . . 4 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
2824, 27eqtr4d 2780 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
2916, 28eqtr4d 2780 . 2 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
307, 29eqtr4d 2780 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  ran crn 5552  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  invcgn 28572   /𝑔 cgs 28573  RingOpscrngo 35789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-1st 7761  df-2nd 7762  df-grpo 28574  df-gid 28575  df-ginv 28576  df-gdiv 28577  df-ablo 28626  df-ass 35738  df-exid 35740  df-mgmOLD 35744  df-sgrOLD 35756  df-mndo 35762  df-rngo 35790
This theorem is referenced by: (None)
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