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Theorem crngm23 37381
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
crngm23 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))

Proof of Theorem crngm23
StepHypRef Expression
1 crngm.1 . . . . 5 𝐺 = (1st𝑅)
2 crngm.2 . . . . 5 𝐻 = (2nd𝑅)
3 crngm.3 . . . . 5 𝑋 = ran 𝐺
41, 2, 3crngocom 37380 . . . 4 ((𝑅 ∈ CRingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
543adant3r1 1179 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
65oveq2d 7420 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐻𝐶)) = (𝐴𝐻(𝐶𝐻𝐵)))
7 crngorngo 37379 . . 3 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
81, 2, 3rngoass 37285 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
97, 8sylan 579 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
101, 2, 3rngoass 37285 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
11103exp2 1351 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐶𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
1211com34 91 . . . 4 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
13123imp2 1346 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
147, 13sylan 579 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
156, 9, 143eqtr4d 2776 1 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  ran crn 5670  cfv 6536  (class class class)co 7404  1st c1st 7969  2nd c2nd 7970  RingOpscrngo 37273  CRingOpsccring 37372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-1st 7971  df-2nd 7972  df-rngo 37274  df-com2 37369  df-crngo 37373
This theorem is referenced by:  crngm4  37382
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