Theorem rngosubdi 36103

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 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Proof of Theorem rngosubdi

Step	Hyp	Ref	Expression
1		ringsubdi.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ringsubdi.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
3		eqid 2738	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
4		ringsubdi.4	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	1, 2, 3, 4	rngosub 36088	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
6	5	3adant3r1 1181	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐷𝐶) = (𝐵𝐺((inv‘𝐺)‘𝐶)))
7	6	oveq2d 7291	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))))
8		ringsubdi.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
9	1, 8, 2	rngocl 36059	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
10	9	3adant3r3 1183	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
11	1, 8, 2	rngocl 36059	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
12	11	3adant3r2 1182	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
13	10, 12	jca 512	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋))
14	1, 2, 3, 4	rngosub 36088	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
15	14	3expb 1119	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋 ∧ (𝐴𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
16	13, 15	syldan 591	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
17		idd 24	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋))
18		idd 24	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → 𝐵 ∈ 𝑋))
19	1, 2, 3	rngonegcl 36085	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘𝐶) ∈ 𝑋)
20	19	ex 413	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐶 ∈ 𝑋 → ((inv‘𝐺)‘𝐶) ∈ 𝑋))
21	17, 18, 20	3anim123d 1442	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)))
22	21	imp 407	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋))
23	1, 8, 2	rngodi 36062	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐶) ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
24	22, 23	syldan 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
25	1, 8, 2, 3	rngonegrmul 36102	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐴𝐻𝐶)) = (𝐴𝐻((inv‘𝐺)‘𝐶)))
26	25	3adant3r2 1182	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐴𝐻𝐶)) = (𝐴𝐻((inv‘𝐺)‘𝐶)))
27	26	oveq2d 7291	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
28	24, 27	eqtr4d 2781	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
29	16, 28	eqtr4d 2781	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))))
30	7, 29	eqtr4d 2781	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ran crn 5590  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  invcgn 28853   /_𝑔 cgs 28854  RingOpscrngo 36052

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ginv 28857  df-gdiv 28858  df-ablo 28907  df-ass 36001  df-exid 36003  df-mgmOLD 36007  df-sgrOLD 36019  df-mndo 36025  df-rngo 36053

This theorem is referenced by:  dmncan1  36234