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Theorem crngm4 33434
Description: Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
crngm.1 𝐺 = (1st𝑅)
crngm.2 𝐻 = (2nd𝑅)
crngm.3 𝑋 = ran 𝐺
Assertion
Ref Expression
crngm4 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Proof of Theorem crngm4
StepHypRef Expression
1 df-3an 1038 . . . . . 6 ((𝐴𝑋𝐵𝑋𝐶𝑋) ↔ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋))
2 crngm.1 . . . . . . 7 𝐺 = (1st𝑅)
3 crngm.2 . . . . . . 7 𝐻 = (2nd𝑅)
4 crngm.3 . . . . . . 7 𝑋 = ran 𝐺
52, 3, 4crngm23 33433 . . . . . 6 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
61, 5sylan2br 493 . . . . 5 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
76adantrrr 760 . . . 4 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
87oveq1d 6619 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9 crngorngo 33431 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
102, 3, 4rngocl 33332 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11103expb 1263 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
1211adantrr 752 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13 simprrl 803 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
14 simprrr 804 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐷𝑋)
1512, 13, 143jca 1240 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋))
162, 3, 4rngoass 33337 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
1715, 16syldan 487 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
189, 17sylan 488 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
192, 3, 4rngocl 33332 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20193expb 1263 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2120adantrlr 758 . . . . . . 7 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2221adantrrr 760 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23 simprlr 802 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
2422, 23, 143jca 1240 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋))
252, 3, 4rngoass 33337 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
2624, 25syldan 487 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
279, 26sylan 488 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
288, 18, 273eqtr3d 2663 . 2 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29283impb 1257 1 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  ran crn 5075  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  RingOpscrngo 33325  CRingOpsccring 33424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-1st 7113  df-2nd 7114  df-rngo 33326  df-com2 33421  df-crngo 33425
This theorem is referenced by:  ispridlc  33501
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