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Theorem maxidlmax 35202
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 35199 . . . . . 6 (𝑅 ∈ RingOps → (𝑀 ∈ (MaxIdl‘𝑅) ↔ (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))))
43biimpa 477 . . . . 5 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → (𝑀 ∈ (Idl‘𝑅) ∧ 𝑀𝑋 ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))))
54simp3d 1136 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))
6 sseq2 3990 . . . . . 6 (𝑗 = 𝐼 → (𝑀𝑗𝑀𝐼))
7 eqeq1 2822 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑀𝐼 = 𝑀))
8 eqeq1 2822 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑋𝐼 = 𝑋))
97, 8orbi12d 912 . . . . . 6 (𝑗 = 𝐼 → ((𝑗 = 𝑀𝑗 = 𝑋) ↔ (𝐼 = 𝑀𝐼 = 𝑋)))
106, 9imbi12d 346 . . . . 5 (𝑗 = 𝐼 → ((𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)) ↔ (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋))))
1110rspcva 3618 . . . 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 459 . 2 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
1413impr 455 1 (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) ∧ (𝐼 ∈ (Idl‘𝑅) ∧ 𝑀𝐼)) → (𝐼 = 𝑀𝐼 = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wss 3933  ran crn 5549  cfv 6348  1st c1st 7676  RingOpscrngo 35053  Idlcidl 35166  MaxIdlcmaxidl 35168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-maxidl 35171
This theorem is referenced by: (None)
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