Theorem crngm4 38542

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 1103	. . . . . 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 38541	. . . . . 6 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
6	1, 5	sylan2br 606	. . . . 5 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
7	6	adantrrr 737	. . . 4 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
8	7	oveq1d 7426	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9		crngorngo 38539	. . . 4 ⊢ (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
10	2, 3, 4	rngocl 38440	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11	10	3expb 1136	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
12	11	adantrr 729	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13		simprrl 792	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐶 ∈ 𝑋)
14		simprrr 793	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐷 ∈ 𝑋)
15	12, 13, 14	3jca 1144	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
16	2, 3, 4	rngoass 38445	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
17	15, 16	syldan 602	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
18	9, 17	sylan 591	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
19	2, 3, 4	rngocl 38440	. . . . . . . . 9 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20	19	3expb 1136	. . . . . . . 8 ⊢ ((𝑅 ∈ RingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
21	20	adantrlr 735	. . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
22	21	adantrrr 737	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23		simprlr 791	. . . . . 6 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → 𝐵 ∈ 𝑋)
24	22, 23, 14	3jca 1144	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))
25	2, 3, 4	rngoass 38445	. . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
26	24, 25	syldan 602	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
27	9, 26	sylan 591	. . 3 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
28	8, 18, 27	3eqtr3d 2812	. 2 ⊢ ((𝑅 ∈ CRingOps ∧ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29	28	3impb 1130	1 ⊢ ((𝑅 ∈ CRingOps ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ran crn 5663  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  RingOpscrngo 38433  CRingOpsccring 38532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7414  df-1st 7986  df-2nd 7987  df-rngo 38434  df-com2 38529  df-crngo 38533

This theorem is referenced by:  ispridlc  38609