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Theorem rngorn1eq 37942
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 = (2nd𝑅)
rnplrnml0.2 𝐺 = (1st𝑅)
Assertion
Ref Expression
rngorn1eq (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

Proof of Theorem rngorn1eq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnplrnml0.2 . . . 4 𝐺 = (1st𝑅)
2 rnplrnml0.1 . . . 4 𝐻 = (2nd𝑅)
3 eqid 2736 . . . 4 ran 𝐺 = ran 𝐺
41, 2, 3rngosm 37908 . . 3 (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
51, 2, 3rngoi 37907 . . . 4 (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
65simprrd 773 . . 3 (𝑅 ∈ RingOps → ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
7 rngmgmbs4 37939 . . 3 ((𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∃𝑥 ∈ ran 𝐺𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) → ran 𝐻 = ran 𝐺)
84, 6, 7syl2anc 584 . 2 (𝑅 ∈ RingOps → ran 𝐻 = ran 𝐺)
98eqcomd 2742 1 (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069   × cxp 5682  ran crn 5685  wf 6556  cfv 6560  (class class class)co 7432  1st c1st 8013  2nd c2nd 8014  AbelOpcablo 30564  RingOpscrngo 37902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-ov 7435  df-1st 8015  df-2nd 8016  df-rngo 37903
This theorem is referenced by:  rngoidmlem  37944  rngo1cl  37947  isdrngo2  37966
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