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Theorem rngorn1eq 37306
Description: In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
Hypotheses
Ref Expression
rnplrnml0.1 𝐻 = (2nd β€˜π‘…)
rnplrnml0.2 𝐺 = (1st β€˜π‘…)
Assertion
Ref Expression
rngorn1eq (𝑅 ∈ RingOps β†’ ran 𝐺 = ran 𝐻)

Proof of Theorem rngorn1eq
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnplrnml0.2 . . . 4 𝐺 = (1st β€˜π‘…)
2 rnplrnml0.1 . . . 4 𝐻 = (2nd β€˜π‘…)
3 eqid 2724 . . . 4 ran 𝐺 = ran 𝐺
41, 2, 3rngosm 37272 . . 3 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
51, 2, 3rngoi 37271 . . . 4 (𝑅 ∈ RingOps β†’ ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺) ∧ (βˆ€π‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran πΊβˆ€π‘§ ∈ ran 𝐺(((π‘₯𝐻𝑦)𝐻𝑧) = (π‘₯𝐻(𝑦𝐻𝑧)) ∧ (π‘₯𝐻(𝑦𝐺𝑧)) = ((π‘₯𝐻𝑦)𝐺(π‘₯𝐻𝑧)) ∧ ((π‘₯𝐺𝑦)𝐻𝑧) = ((π‘₯𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))))
65simprrd 771 . . 3 (𝑅 ∈ RingOps β†’ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
7 rngmgmbs4 37303 . . 3 ((𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦)) β†’ ran 𝐻 = ran 𝐺)
84, 6, 7syl2anc 583 . 2 (𝑅 ∈ RingOps β†’ ran 𝐻 = ran 𝐺)
98eqcomd 2730 1 (𝑅 ∈ RingOps β†’ ran 𝐺 = ran 𝐻)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   Γ— cxp 5665  ran crn 5668  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402  1st c1st 7967  2nd c2nd 7968  AbelOpcablo 30292  RingOpscrngo 37266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-ov 7405  df-1st 7969  df-2nd 7970  df-rngo 37267
This theorem is referenced by:  rngoidmlem  37308  rngo1cl  37311  isdrngo2  37330
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