Theorem rngosubdi 36259

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 2736	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
4		ringsubdi.4	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	1, 2, 3, 4	rngosub 36244	. . . 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 7354	. 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 36215	. . . . . 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 36215	. . . . . 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 36244	. . . . 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 36241	. . . . . . . 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 36218	. . . . 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 36258	. . . . . 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 7354	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻((inv‘𝐺)‘𝐶))))
28	24, 27	eqtr4d 2779	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))) = ((𝐴𝐻𝐵)𝐺((inv‘𝐺)‘(𝐴𝐻𝐶))))
29	16, 28	eqtr4d 2779	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)) = (𝐴𝐻(𝐵𝐺((inv‘𝐺)‘𝐶))))
30	7, 29	eqtr4d 2779	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ran crn 5622  ‘cfv 6480  (class class class)co 7338  1^st c1st 7898  2^nd c2nd 7899  invcgn 29142   /_𝑔 cgs 29143  RingOpscrngo 36208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4854  df-iun 4944  df-br 5094  df-opab 5156  df-mpt 5177  df-id 5519  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-1st 7900  df-2nd 7901  df-grpo 29144  df-gid 29145  df-ginv 29146  df-gdiv 29147  df-ablo 29196  df-ass 36157  df-exid 36159  df-mgmOLD 36163  df-sgrOLD 36175  df-mndo 36181  df-rngo 36209

This theorem is referenced by:  dmncan1  36390