Theorem crngm4 35283

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 1085	. . . . . 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 35282	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
6	1, 5	sylan2br 596	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
7	6	adantrrr 723	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
8	7	oveq1d 7173	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9		crngorngo 35280	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
10	2, 3, 4	rngocl 35181	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	3expb 1116	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
12	11	adantrr 715	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13		simprrl 779	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐶 ∈ 𝑋)
14		simprrr 780	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐷 ∈ 𝑋)
15	12, 13, 14	3jca 1124	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
16	2, 3, 4	rngoass 35186	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
17	15, 16	syldan 593	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
18	9, 17	sylan 582	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
19	2, 3, 4	rngocl 35181	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20	19	3expb 1116	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
21	20	adantrlr 721	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
22	21	adantrrr 723	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23		simprlr 778	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐵 ∈ 𝑋)
24	22, 23, 14	3jca 1124	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
25	2, 3, 4	rngoass 35186	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
26	24, 25	syldan 593	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
27	9, 26	sylan 582	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
28	8, 18, 27	3eqtr3d 2866	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29	28	3impb 1111	1 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ran crn 5558  ‘cfv 6357  (class class class)co 7158  1^st c1st 7689  2^nd c2nd 7690  RingOpscrngo 35174  CRingOpsccring 35273

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-1st 7691  df-2nd 7692  df-rngo 35175  df-com2 35270  df-crngo 35274

This theorem is referenced by:  ispridlc  35350