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Theorem crngm23 34719
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 34718 . . . 4 ((𝑅 ∈ CRingOps ∧ 𝐵𝑋𝐶𝑋) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
543adant3r1 1162 . . 3 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐵𝐻𝐶) = (𝐶𝐻𝐵))
65oveq2d 6992 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐻𝐶)) = (𝐴𝐻(𝐶𝐻𝐵)))
7 crngorngo 34717 . . 3 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
81, 2, 3rngoass 34623 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
97, 8sylan 572 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
101, 2, 3rngoass 34623 . . . . . 6 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐶𝑋𝐵𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
11103exp2 1334 . . . . 5 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐶𝑋 → (𝐵𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
1211com34 91 . . . 4 (𝑅 ∈ RingOps → (𝐴𝑋 → (𝐵𝑋 → (𝐶𝑋 → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵))))))
13123imp2 1329 . . 3 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
147, 13sylan 572 . 2 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐶)𝐻𝐵) = (𝐴𝐻(𝐶𝐻𝐵)))
156, 9, 143eqtr4d 2825 1 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  w3a 1068   = wceq 1507  wcel 2050  ran crn 5408  cfv 6188  (class class class)co 6976  1st c1st 7499  2nd c2nd 7500  RingOpscrngo 34611  CRingOpsccring 34710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-1st 7501  df-2nd 7502  df-rngo 34612  df-com2 34707  df-crngo 34711
This theorem is referenced by:  crngm4  34720
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