Theorem rngosubdir 37906

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 2740	. . . . 5 ⊢ (inv‘𝐺) = (inv‘𝐺)
4		ringsubdi.4	. . . . 5 ⊢ 𝐷 = ( /𝑔 ‘𝐺)
5	1, 2, 3, 4	rngosub 37890	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
6	5	3adant3r3 1184	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) = (𝐴𝐺((inv‘𝐺)‘𝐵)))
7	6	oveq1d 7463	. 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 37861	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
10	9	3adant3r2 1183	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
11	1, 8, 2	rngocl 37861	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) ∈ 𝑋)
12	11	3adant3r1 1182	. . . . 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 37890	. . . . 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 590	. . 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 37887	. . . . . . . 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 1443	. . . . . 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 37865	. . . . 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 590	. . . 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 37903	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
26	25	3adant3r1 1182	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((inv‘𝐺)‘(𝐵𝐻𝐶)) = (((inv‘𝐺)‘𝐵)𝐻𝐶))
27	26	oveq2d 7464	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))) = ((𝐴𝐻𝐶)𝐺(((inv‘𝐺)‘𝐵)𝐻𝐶)))
28	24, 27	eqtr4d 2783	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶) = ((𝐴𝐻𝐶)𝐺((inv‘𝐺)‘(𝐵𝐻𝐶))))
29	16, 28	eqtr4d 2783	. 2 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)) = ((𝐴𝐺((inv‘𝐺)‘𝐵))𝐻𝐶))
30	7, 29	eqtr4d 2783	1 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  invcgn 30523   /_𝑔 cgs 30524  RingOpscrngo 37854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-grpo 30525  df-gid 30526  df-ginv 30527  df-gdiv 30528  df-ablo 30577  df-ass 37803  df-exid 37805  df-mgmOLD 37809  df-sgrOLD 37821  df-mndo 37827  df-rngo 37855

This theorem is referenced by: (None)