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Theorem maxidlmax 38005
Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
Hypotheses
Ref Expression
maxidlnr.1 𝐺 = (1st𝑅)
maxidlnr.2 𝑋 = ran 𝐺
Assertion
Ref Expression
maxidlmax (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))

Proof of Theorem maxidlmax
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 maxidlnr.1 . . . . . . 7 𝐺 = (1st𝑅)
2 maxidlnr.2 . . . . . . 7 𝑋 = ran 𝐺
31, 2ismaxidl 38002 . . . . . 6 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
43biimpa 476 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
54simp3d 1144 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))
6 sseq2 4035 . . . . . 6 (𝑗 = 𝐼 → (𝑀𝑗𝑀𝐼))
7 eqeq1 2744 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑀𝐼 = 𝑀))
8 eqeq1 2744 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑋𝐼 = 𝑋))
97, 8orbi12d 917 . . . . . 6 (𝑗 = 𝐼 → ((𝑗 = 𝑀𝑗 = 𝑋) ↔ (𝐼 = 𝑀𝐼 = 𝑋)))
106, 9imbi12d 344 . . . . 5 (𝑗 = 𝐼 → ((𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)) ↔ (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋))))
1110rspcva 3633 . . . 4 ((𝐼 ∈ (Idl‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
125, 11sylan2 592 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ (𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅))) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
1312ancoms 458 . 2 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
1413impr 454 1 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1087   = wceq 1537  wcel 2108  wne 2946  wral 3067  wss 3976  ran crn 5701  cfv 6575  1st c1st 8030  RingOpscrngo 37856  Idlcidl 37969  MaxIdlcmaxidl 37971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6527  df-fun 6577  df-fv 6583  df-maxidl 37974
This theorem is referenced by: (None)
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