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Theorem maxidlnr 33011
 Description: A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
maxidlnr.1 𝐺 = (1st𝑅)
maxidlnr.2 𝑋 = ran 𝐺
Assertion
Ref Expression
maxidlnr ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀𝑋)

Proof of Theorem maxidlnr
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 maxidlnr.1 . . . 4 𝐺 = (1st𝑅)
2 maxidlnr.2 . . . 4 𝑋 = ran 𝐺
31, 2ismaxidl 33009 . . 3 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
43biimpa 500 . 2 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
54simp2d 1067 1 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → 𝑀𝑋)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ran crn 5039  ‘cfv 5804  1st c1st 7057  RingOpscrngo 32863  Idlcidl 32976  MaxIdlcmaxidl 32978 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-maxidl 32981 This theorem is referenced by:  maxidln1  33013
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