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Theorem rngoass 36415
Description: Associative law for the multiplication operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ringi.1 𝐺 = (1st𝑅)
ringi.2 𝐻 = (2nd𝑅)
ringi.3 𝑋 = ran 𝐺
Assertion
Ref Expression
rngoass ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))

Proof of Theorem rngoass
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . . 6 𝐺 = (1st𝑅)
2 ringi.2 . . . . . 6 𝐻 = (2nd𝑅)
3 ringi.3 . . . . . 6 𝑋 = ran 𝐺
41, 2, 3rngoi 36408 . . . . 5 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
54simprd 497 . . . 4 (𝑅 ∈ RingOps → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
65simpld 496 . . 3 (𝑅 ∈ RingOps → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
7 simp1 1137 . . . . 5 ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
87ralimi 3083 . . . 4 (∀𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
982ralimi 3123 . . 3 (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
106, 9syl 17 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
11 oveq1 7368 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
1211oveq1d 7376 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝑦)𝐻𝑧))
13 oveq1 7368 . . . 4 (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝑦𝐻𝑧)))
1412, 13eqeq12d 2749 . . 3 (𝑥 = 𝐴 → (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧))))
15 oveq2 7369 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
1615oveq1d 7376 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝑧))
17 oveq1 7368 . . . . 5 (𝑦 = 𝐵 → (𝑦𝐻𝑧) = (𝐵𝐻𝑧))
1817oveq2d 7377 . . . 4 (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝑧)))
1916, 18eqeq12d 2749 . . 3 (𝑦 = 𝐵 → (((𝐴𝐻𝑦)𝐻𝑧) = (𝐴𝐻(𝑦𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧))))
20 oveq2 7369 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐻𝑧) = ((𝐴𝐻𝐵)𝐻𝐶))
21 oveq2 7369 . . . . 5 (𝑧 = 𝐶 → (𝐵𝐻𝑧) = (𝐵𝐻𝐶))
2221oveq2d 7377 . . . 4 (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐻𝑧)) = (𝐴𝐻(𝐵𝐻𝐶)))
2320, 22eqeq12d 2749 . . 3 (𝑧 = 𝐶 → (((𝐴𝐻𝐵)𝐻𝑧) = (𝐴𝐻(𝐵𝐻𝑧)) ↔ ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))))
2414, 19, 23rspc3v 3595 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶))))
2510, 24mpan9 508 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = (𝐴𝐻(𝐵𝐻𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3061  wrex 3070   × cxp 5635  ran crn 5638  wf 6496  cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  AbelOpcablo 29535  RingOpscrngo 36403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7364  df-1st 7925  df-2nd 7926  df-rngo 36404
This theorem is referenced by:  rngomndo  36444  rngoneglmul  36452  rngonegrmul  36453  zerdivemp1x  36456  isdrngo2  36467  crngm23  36511  crngm4  36512  prnc  36576
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