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Theorem rngodi 35685
Description: Distributive law for the multiplication operation of a ring (left-distributivity). (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
rngodi ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))

Proof of Theorem rngodi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . . 5 𝐺 = (1st𝑅)
2 ringi.2 . . . . 5 𝐻 = (2nd𝑅)
3 ringi.3 . . . . 5 𝑋 = ran 𝐺
41, 2, 3rngoi 35680 . . . 4 (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 499 . . 3 (𝑅 ∈ RingOps → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
65simpld 498 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
7 simp2 1138 . . . . 5 ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
87ralimi 3075 . . . 4 (∀𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
982ralimi 3076 . . 3 (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
10 oveq1 7177 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝑦𝐺𝑧)))
11 oveq1 7177 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
12 oveq1 7177 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
1311, 12oveq12d 7188 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)))
1410, 13eqeq12d 2754 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ↔ (𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧))))
15 oveq1 7177 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
1615oveq2d 7186 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝑧)))
17 oveq2 7178 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
1817oveq1d 7185 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)))
1916, 18eqeq12d 2754 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧))))
20 oveq2 7178 . . . . . 6 (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
2120oveq2d 7186 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝐶)))
22 oveq2 7178 . . . . . 6 (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶))
2322oveq2d 7186 . . . . 5 (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
2421, 23eqeq12d 2754 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
2514, 19, 24rspc3v 3539 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
269, 25syl5 34 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
276, 26mpan9 510 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1088   = wceq 1542  wcel 2114  wral 3053  wrex 3054   × cxp 5523  ran crn 5526  wf 6335  cfv 6339  (class class class)co 7170  1st c1st 7712  2nd c2nd 7713  AbelOpcablo 28479  RingOpscrngo 35675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-1st 7714  df-2nd 7715  df-rngo 35676
This theorem is referenced by:  rngorz  35704  rngonegmn1r  35723  rngosubdi  35726
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