Theorem crngm4 38143

Description: Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)

Hypotheses

Ref	Expression
crngm.1	⊢ 𝐺 = (1st ‘𝑅)
crngm.2	⊢ 𝐻 = (2nd ‘𝑅)
crngm.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
crngm4	⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Proof of Theorem crngm4

Step	Hyp	Ref	Expression
1		df-3an 1088	. . . . . 6 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) ↔ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋))
2		crngm.1	. . . . . . 7 ⊢ 𝐺 = (1st ‘𝑅)
3		crngm.2	. . . . . . 7 ⊢ 𝐻 = (2nd ‘𝑅)
4		crngm.3	. . . . . . 7 ⊢ 𝑋 = ran 𝐺
5	2, 3, 4	crngm23 38142	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
6	1, 5	sylan2br 595	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
7	6	adantrrr 725	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
8	7	oveq1d 7371	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9		crngorngo 38140	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
10	2, 3, 4	rngocl 38041	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	3expb 1120	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
12	11	adantrr 717	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13		simprrl 780	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐶 ∈ 𝑋)
14		simprrr 781	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐷 ∈ 𝑋)
15	12, 13, 14	3jca 1128	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
16	2, 3, 4	rngoass 38046	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
17	15, 16	syldan 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
18	9, 17	sylan 580	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
19	2, 3, 4	rngocl 38041	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20	19	3expb 1120	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
21	20	adantrlr 723	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
22	21	adantrrr 725	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23		simprlr 779	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐵 ∈ 𝑋)
24	22, 23, 14	3jca 1128	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
25	2, 3, 4	rngoass 38046	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
26	24, 25	syldan 591	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
27	9, 26	sylan 580	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
28	8, 18, 27	3eqtr3d 2777	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29	28	3impb 1114	1 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ran crn 5623  ‘cfv 6490  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  RingOpscrngo 38034  CRingOpsccring 38133

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-ov 7359  df-1st 7931  df-2nd 7932  df-rngo 38035  df-com2 38130  df-crngo 38134

This theorem is referenced by:  ispridlc  38210