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Theorem crngm4 34134
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 1073 . . . . . 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 34133 . . . . . 6 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
61, 5sylan2br 582 . . . . 5 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
76adantrrr 704 . . . 4 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))
87oveq1d 6811 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷))
9 crngorngo 34131 . . . 4 (𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)
102, 3, 4rngocl 34032 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) ∈ 𝑋)
11103expb 1113 . . . . . . 7 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝑋)
1211adantrr 696 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐵) ∈ 𝑋)
13 simprrl 766 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐶𝑋)
14 simprrr 767 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐷𝑋)
1512, 13, 143jca 1122 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋))
162, 3, 4rngoass 34037 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐵) ∈ 𝑋𝐶𝑋𝐷𝑋)) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
1715, 16syldan 579 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
189, 17sylan 569 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐵)𝐻𝐶)𝐻𝐷) = ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)))
192, 3, 4rngocl 34032 . . . . . . . . 9 ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐶𝑋) → (𝐴𝐻𝐶) ∈ 𝑋)
20193expb 1113 . . . . . . . 8 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2120adantrlr 702 . . . . . . 7 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ 𝐶𝑋)) → (𝐴𝐻𝐶) ∈ 𝑋)
2221adantrrr 704 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (𝐴𝐻𝐶) ∈ 𝑋)
23 simprlr 765 . . . . . 6 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → 𝐵𝑋)
2422, 23, 143jca 1122 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋))
252, 3, 4rngoass 34037 . . . . 5 ((𝑅 ∈ RingOps ∧ ((𝐴𝐻𝐶) ∈ 𝑋𝐵𝑋𝐷𝑋)) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
2624, 25syldan 579 . . . 4 ((𝑅 ∈ RingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
279, 26sylan 569 . . 3 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → (((𝐴𝐻𝐶)𝐻𝐵)𝐻𝐷) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
288, 18, 273eqtr3d 2813 . 2 ((𝑅 ∈ CRingOps ∧ ((𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋))) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
29283impb 1107 1 ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  ran crn 5251  cfv 6030  (class class class)co 6796  1st c1st 7317  2nd c2nd 7318  RingOpscrngo 34025  CRingOpsccring 34124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-1st 7319  df-2nd 7320  df-rngo 34026  df-com2 34121  df-crngo 34125
This theorem is referenced by:  ispridlc  34201
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