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Theorem rngosm 36058
Description: Functionality of 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
rngosm (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)

Proof of Theorem rngosm
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ringi.1 . . . 4 𝐺 = (1st𝑅)
2 ringi.2 . . . 4 𝐻 = (2nd𝑅)
3 ringi.3 . . . 4 𝑋 = ran 𝐺
41, 2, 3rngoi 36057 . . 3 (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))))
54simpld 495 . 2 (𝑅 ∈ RingOps → (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋))
65simprd 496 1 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065   × cxp 5587  ran crn 5590  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  AbelOpcablo 28906  RingOpscrngo 36052
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-1st 7831  df-2nd 7832  df-rngo 36053
This theorem is referenced by:  rngocl  36059  rngosn3  36082  rngodm1dm2  36090  rngorn1eq  36092  rngomndo  36093  divrngcl  36115  isdrngo2  36116
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