Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crngm4 Structured version   Visualization version   GIF version

Theorem crngm4 34343
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 1113 . . . . . 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 34342 . . . . . 6 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
61, 5sylan2br 588 . . . . 5 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
76adantrrr 716 . . . 4 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
87oveq1d 6925 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9 crngorngo 34340 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
102, 3, 4rngocl 34241 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11103expb 1153 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
1211adantrr 708 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13 simprrl 799 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
14 simprrr 800 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐷𝑋)
1512, 13, 143jca 1162 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋))
162, 3, 4rngoass 34246 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
1715, 16syldan 585 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
189, 17sylan 575 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
192, 3, 4rngocl 34241 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20193expb 1153 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2120adantrlr 714 . . . . . . 7 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2221adantrrr 716 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23 simprlr 798 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
2422, 23, 143jca 1162 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋))
252, 3, 4rngoass 34246 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
2624, 25syldan 585 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
279, 26sylan 575 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
288, 18, 273eqtr3d 2869 . 2 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29283impb 1147 1 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1111   = wceq 1656  wcel 2164  ran crn 5347  cfv 6127  (class class class)co 6910  1st c1st 7431  2nd c2nd 7432  RingOpscrngo 34234  CRingOpsccring 34333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-ov 6913  df-1st 7433  df-2nd 7434  df-rngo 34235  df-com2 34330  df-crngo 34334
This theorem is referenced by:  ispridlc  34410
  Copyright terms: Public domain W3C validator