Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  maxidlmax Structured version   Visualization version   GIF version

Theorem maxidlmax 37744
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 37741 . . . . . 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 4006 . . . . . 6 (𝑗 = 𝐼 → (𝑀𝑗𝑀𝐼))
7 eqeq1 2730 . . . . . . 7 (𝑗 = 𝐼 → (𝑗 = 𝑀𝐼 = 𝑀))
8 eqeq1 2730 . . . . . . 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 3606 . . . 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 1534  wcel 2099  wne 2930  wral 3051  wss 3947  ran crn 5683  cfv 6554  1st c1st 8001  RingOpscrngo 37595  Idlcidl 37708  MaxIdlcmaxidl 37710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fv 6562  df-maxidl 37713
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator