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Theorem rngodi 33332
 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 33327 . . . 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 479 . . 3 (𝑅 ∈ RingOps → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥𝑋𝑦𝑋 ((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
65simpld 475 . 2 (𝑅 ∈ RingOps → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))))
7 simp2 1060 . . . . . 6 ((((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
87ralimi 2947 . . . . 5 (∀𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
98ralimi 2947 . . . 4 (∀𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑦𝑋𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
109ralimi 2947 . . 3 (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → ∀𝑥𝑋𝑦𝑋𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))
11 oveq1 6611 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝑦𝐺𝑧)))
12 oveq1 6611 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐻𝑦) = (𝐴𝐻𝑦))
13 oveq1 6611 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝐻𝑧) = (𝐴𝐻𝑧))
1412, 13oveq12d 6622 . . . . 5 (𝑥 = 𝐴 → ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)))
1511, 14eqeq12d 2636 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ↔ (𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧))))
16 oveq1 6611 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐺𝑧) = (𝐵𝐺𝑧))
1716oveq2d 6620 . . . . 5 (𝑦 = 𝐵 → (𝐴𝐻(𝑦𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝑧)))
18 oveq2 6612 . . . . . 6 (𝑦 = 𝐵 → (𝐴𝐻𝑦) = (𝐴𝐻𝐵))
1918oveq1d 6619 . . . . 5 (𝑦 = 𝐵 → ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)))
2017, 19eqeq12d 2636 . . . 4 (𝑦 = 𝐵 → ((𝐴𝐻(𝑦𝐺𝑧)) = ((𝐴𝐻𝑦)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧))))
21 oveq2 6612 . . . . . 6 (𝑧 = 𝐶 → (𝐵𝐺𝑧) = (𝐵𝐺𝐶))
2221oveq2d 6620 . . . . 5 (𝑧 = 𝐶 → (𝐴𝐻(𝐵𝐺𝑧)) = (𝐴𝐻(𝐵𝐺𝐶)))
23 oveq2 6612 . . . . . 6 (𝑧 = 𝐶 → (𝐴𝐻𝑧) = (𝐴𝐻𝐶))
2423oveq2d 6620 . . . . 5 (𝑧 = 𝐶 → ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
2522, 24eqeq12d 2636 . . . 4 (𝑧 = 𝐶 → ((𝐴𝐻(𝐵𝐺𝑧)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝑧)) ↔ (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
2615, 20, 25rspc3v 3309 . . 3 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
2710, 26syl5 34 . 2 ((𝐴𝑋𝐵𝑋𝐶𝑋) → (∀𝑥𝑋𝑦𝑋𝑧𝑋 (((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶))))
286, 27mpan9 486 1 ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐺𝐶)) = ((𝐴𝐻𝐵)𝐺(𝐴𝐻𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908   × cxp 5072  ran crn 5075  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  AbelOpcablo 27244  RingOpscrngo 33322 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-1st 7113  df-2nd 7114  df-rngo 33323 This theorem is referenced by:  rngorz  33351  rngonegmn1r  33370  rngosubdi  33373
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