Theorem crngm23 38142

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
crngm23	⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))

Proof of Theorem crngm23

Step	Hyp	Ref	Expression
1		crngm.1	. . . . 5 ⊢ 𝐺 = (1st ‘𝑅)
2		crngm.2	. . . . 5 ⊢ 𝐻 = (2nd ‘𝑅)
3		crngm.3	. . . . 5 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	crngocom 38141	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
5	4	3adant3r1 1183	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
6	5	oveq2d 7372	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻(𝐵𝐻𝐶)) = (𝐴𝐻(𝐶𝐻𝐵)))
7		crngorngo 38140	. . 3 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
8	1, 2, 3	rngoass 38046	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
9	7, 8	sylan 580	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
10	1, 2, 3	rngoass 38046	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
11	10	3exp2 1355	. . . . 5 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐶 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
12	11	com34 91	. . . 4 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐶 ∈ 𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
13	12	3imp2 1350	. . 3 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
14	7, 13	sylan 580	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
15	6, 9, 14	3eqtr4d 2779	1 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))

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:  crngm4  38143