Theorem rngosubdir 37926

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
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 2729	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
4		ringsubdi.4	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	1, 2, 3, 4	rngosub 37910	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
6	5	3adant3r3 1185	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
7	6	oveq1d 7364	. 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 37881	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
10	9	3adant3r2 1184	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
11	1, 8, 2	rngocl 37881	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) ∈ 𝑋)
12	11	3adant3r1 1183	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) ∈ 𝑋)
13	10, 12	jca 511	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋))
14	1, 2, 3, 4	rngosub 37910	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
15	14	3expb 1120	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ (𝐵𝐻𝐶) ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
16	13, 15	syldan 591	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
17		idd 24	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑋))
18	1, 2, 3	rngonegcl 37907	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋) → ((inv‘𝐺)‘𝐵) ∈ 𝑋)
19	18	ex 412	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐵 ∈ 𝑋 → ((inv‘𝐺)‘𝐵) ∈ 𝑋))
20		idd 24	. . . . . . 7 ⊢ (𝑅 ∈ RingOps → (𝐶 ∈ 𝑋 → 𝐶 ∈ 𝑋))
21	17, 19, 20	3anim123d 1445	. . . . . 6 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)))
22	21	imp 406	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴 ∈ 𝑋 ∧ ((inv‘𝐺)‘𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋))
23	1, 8, 2	rngodir 37885	. . . . 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 591	. . . 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 37923	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
26	25	3adant3r1 1183	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
27	26	oveq2d 7365	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
28	24, 27	eqtr4d 2767	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
29	16, 28	eqtr4d 2767	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
30	7, 29	eqtr4d 2767	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ran crn 5620  ‘cfv 6482  (class class class)co 7349  1^st c1st 7922  2^nd c2nd 7923  invcgn 30435   /_𝑔 cgs 30436  RingOpscrngo 37874

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-grpo 30437  df-gid 30438  df-ginv 30439  df-gdiv 30440  df-ablo 30489  df-ass 37823  df-exid 37825  df-mgmOLD 37829  df-sgrOLD 37841  df-mndo 37847  df-rngo 37875

This theorem is referenced by: (None)