Theorem rngosubdir 38657

Description: Obsolete theorem, use ringsubdir 20439 instead. Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

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
rngosubdir	⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Proof of Theorem rngosubdir

Step	Hyp	Ref	Expression
1		ringsubdi.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		ringsubdi.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
3		eqid 2765	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
4		ringsubdi.4	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	1, 2, 3, 4	rngosub 38641	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
6	5	3adant3r3 1203	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
7	6	oveq1d 7434	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
8		ringsubdi.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
9	1, 8, 2	rngocl 38612	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
10	9	3adant3r2 1202	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
11	1, 8, 2	rngocl 38612	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) ∈ 𝑋)
12	11	3adant3r1 1201	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) ∈ 𝑋)
13	10, 12	jca 521	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋))
14	1, 2, 3, 4	rngosub 38641	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
15	14	3expb 1138	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
16	13, 15	syldan 603	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
17		idd 25	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋))
18	1, 2, 3	rngonegcl 38638	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
19	18	ex 418	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((inv‘𝐺)‘𝐵) ∈ 𝑋))
20		idd 25	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐶 ∈ 𝑋 → 𝐶 ∈ 𝑋))
21	17, 19, 20	3anim123d 1471	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)))
22	21	imp 412	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
23	1, 8, 2	rngodir 38616	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
24	22, 23	syldan 603	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
25	1, 8, 2, 3	rngoneglmul 38654	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
26	25	3adant3r1 1201	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
27	26	oveq2d 7435	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
28	24, 27	eqtr4d 2803	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
29	16, 28	eqtr4d 2803	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
30	7, 29	eqtr4d 2803	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ran crn 5664  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  invcgn 30916   /_𝑔 cgs 30917  RingOpscrngo 38605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-grpo 30918  df-gid 30919  df-ginv 30920  df-gdiv 30921  df-ablo 30970  df-ass 38554  df-exid 38556  df-mgmOLD 38560  df-sgrOLD 38572  df-mndo 38578  df-rngo 38606

This theorem is used by: (None)