Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mxidlmax Structured version   Visualization version   GIF version

Theorem mxidlmax 32289
Description: A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
Hypothesis
Ref Expression
mxidlval.1 𝐡 = (Baseβ€˜π‘…)
Assertion
Ref Expression
mxidlmax (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝐼)) β†’ (𝐼 = 𝑀 ∨ 𝐼 = 𝐡))

Proof of Theorem mxidlmax
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 sseq2 3974 . . . 4 (𝑗 = 𝐼 β†’ (𝑀 βŠ† 𝑗 ↔ 𝑀 βŠ† 𝐼))
2 eqeq1 2737 . . . . 5 (𝑗 = 𝐼 β†’ (𝑗 = 𝑀 ↔ 𝐼 = 𝑀))
3 eqeq1 2737 . . . . 5 (𝑗 = 𝐼 β†’ (𝑗 = 𝐡 ↔ 𝐼 = 𝐡))
42, 3orbi12d 918 . . . 4 (𝑗 = 𝐼 β†’ ((𝑗 = 𝑀 ∨ 𝑗 = 𝐡) ↔ (𝐼 = 𝑀 ∨ 𝐼 = 𝐡)))
51, 4imbi12d 345 . . 3 (𝑗 = 𝐼 β†’ ((𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)) ↔ (𝑀 βŠ† 𝐼 β†’ (𝐼 = 𝑀 ∨ 𝐼 = 𝐡))))
6 mxidlval.1 . . . . . . 7 𝐡 = (Baseβ€˜π‘…)
76ismxidl 32286 . . . . . 6 (𝑅 ∈ Ring β†’ (𝑀 ∈ (MaxIdealβ€˜π‘…) ↔ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))))
87biimpa 478 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ (𝑀 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 β‰  𝐡 ∧ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡))))
98simp3d 1145 . . . 4 ((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) β†’ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))
109adantr 482 . . 3 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝐼 ∈ (LIdealβ€˜π‘…)) β†’ βˆ€π‘— ∈ (LIdealβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝐡)))
11 simpr 486 . . 3 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝐼 ∈ (LIdealβ€˜π‘…)) β†’ 𝐼 ∈ (LIdealβ€˜π‘…))
125, 10, 11rspcdva 3584 . 2 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ 𝐼 ∈ (LIdealβ€˜π‘…)) β†’ (𝑀 βŠ† 𝐼 β†’ (𝐼 = 𝑀 ∨ 𝐼 = 𝐡)))
1312impr 456 1 (((𝑅 ∈ Ring ∧ 𝑀 ∈ (MaxIdealβ€˜π‘…)) ∧ (𝐼 ∈ (LIdealβ€˜π‘…) ∧ 𝑀 βŠ† 𝐼)) β†’ (𝐼 = 𝑀 ∨ 𝐼 = 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3914  β€˜cfv 6500  Basecbs 17091  Ringcrg 19972  LIdealclidl 20676  MaxIdealcmxidl 32283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-mxidl 32284
This theorem is referenced by:  mxidlprm  32292  zarclssn  32518
  Copyright terms: Public domain W3C validator