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Theorem rngorn1eq 37407
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 2728 . . . 4 ran 𝐺 = ran 𝐺
41, 2, 3rngosm 37373 . . 3 (𝑅 ∈ RingOps β†’ 𝐻:(ran 𝐺 Γ— ran 𝐺)⟢ran 𝐺)
51, 2, 3rngoi 37372 . . . 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 773 . . 3 (𝑅 ∈ RingOps β†’ βˆƒπ‘₯ ∈ ran πΊβˆ€π‘¦ ∈ ran 𝐺((π‘₯𝐻𝑦) = 𝑦 ∧ (𝑦𝐻π‘₯) = 𝑦))
7 rngmgmbs4 37404 . . 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 2734 1 (𝑅 ∈ RingOps β†’ ran 𝐺 = ran 𝐻)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  βˆ€wral 3058  βˆƒwrex 3067   Γ— cxp 5676  ran crn 5679  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  AbelOpcablo 30367  RingOpscrngo 37367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fo 6554  df-fv 6556  df-ov 7423  df-1st 7993  df-2nd 7994  df-rngo 37368
This theorem is referenced by:  rngoidmlem  37409  rngo1cl  37412  isdrngo2  37431
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