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Theorem maxidlmax 37573
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 37570 . . . . . 6 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
43biimpa 475 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
54simp3d 1141 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))
6 sseq2 3999 . . . . . 6 (𝑗 = 𝐼 → (𝑀𝑗𝑀𝐼))
7 eqeq1 2729 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑀𝐼 = 𝑀))
8 eqeq1 2729 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑋𝐼 = 𝑋))
97, 8orbi12d 916 . . . . . 6 (𝑗 = 𝐼 → ((𝑗 = 𝑀𝑗 = 𝑋) ↔ (𝐼 = 𝑀𝐼 = 𝑋)))
106, 9imbi12d 343 . . . . 5 (𝑗 = 𝐼 → ((𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)) ↔ (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋))))
1110rspcva 3599 . . . 4 ((𝐼 ∈ (Idl‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
125, 11sylan2 591 . . 3 ((𝐼 ∈ (Idl‘𝑅) ∧ (𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅))) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
1312ancoms 457 . 2 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
1413impr 453 1 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wne 2930  wral 3051  wss 3939  ran crn 5673  cfv 6543  1st c1st 7989  RingOpscrngo 37424  Idlcidl 37537  MaxIdlcmaxidl 37539
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6495  df-fun 6545  df-fv 6551  df-maxidl 37542
This theorem is referenced by: (None)
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