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Theorem maxidlmax 38044
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 38041 . . . . . 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 1145 . . . 4 ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdl‘𝑅)) → ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)))
6 sseq2 4025 . . . . . 6 (𝑗 = 𝐼 → (𝑀𝑗𝑀𝐼))
7 eqeq1 2741 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑀𝐼 = 𝑀))
8 eqeq1 2741 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑋𝐼 = 𝑋))
97, 8orbi12d 919 . . . . . 6 (𝑗 = 𝐼 → ((𝑗 = 𝑀𝑗 = 𝑋) ↔ (𝐼 = 𝑀𝐼 = 𝑋)))
106, 9imbi12d 344 . . . . 5 (𝑗 = 𝐼 → ((𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋)) ↔ (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋))))
1110rspcva 3623 . . . 4 ((𝐼 ∈ (Idl‘𝑅) ∧ ∀𝑗 ∈ (Idl‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝑋))) → (𝑀𝐼 → (𝐼 = 𝑀𝐼 = 𝑋)))
125, 11sylan2 593 . . 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 848  w3a 1087   = wceq 1539  wcel 2108  wne 2940  wral 3061  wss 3966  ran crn 5694  cfv 6569  1st c1st 8020  RingOpscrngo 37895  Idlcidl 38008  MaxIdlcmaxidl 38010
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-opab 5214  df-mpt 5235  df-id 5587  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-iota 6522  df-fun 6571  df-fv 6577  df-maxidl 38013
This theorem is referenced by: (None)
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