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Theorem rngosm 34656
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 34655 . . 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 1078   = wceq 1520  wcel 2079  wral 3103  wrex 3104   × cxp 5433  ran crn 5436  wf 6213  cfv 6217  (class class class)co 7007  1st c1st 7534  2nd c2nd 7535  AbelOpcablo 28000  RingOpscrngo 34650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225  df-ov 7010  df-1st 7536  df-2nd 7537  df-rngo 34651
This theorem is referenced by:  rngocl  34657  rngosn3  34680  rngodm1dm2  34688  rngorn1eq  34690  rngomndo  34691  divrngcl  34713  isdrngo2  34714
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