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Theorem rngosm 33981
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 33980 . . 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 477 . 2 (𝑅 ∈ RingOps → (𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋))
65simprd 482 1 (𝑅 ∈ RingOps → 𝐻:(𝑋 × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1620  wcel 2127  wral 3038  wrex 3039   × cxp 5252  ran crn 5255  wf 6033  cfv 6037  (class class class)co 6801  1st c1st 7319  2nd c2nd 7320  AbelOpcablo 27678  RingOpscrngo 33975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-fv 6045  df-ov 6804  df-1st 7321  df-2nd 7322  df-rngo 33976
This theorem is referenced by:  rngocl  33982  rngosn3  34005  rngodm1dm2  34013  rngorn1eq  34015  rngomndo  34016  divrngcl  34038  isdrngo2  34039
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