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Theorem ismxidl 30978
Description: The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.) (Revised by Thierry Arnoux, 19-Jan-2024.)
Hypothesis
Ref Expression
mxidlval.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
ismxidl (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
Distinct variable groups:   𝑅,𝑗   𝑗,𝑀
Allowed substitution hint:   𝐵(𝑗)

Proof of Theorem ismxidl
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 mxidlval.1 . . . 4 𝐵 = (Base‘𝑅)
21mxidlval 30977 . . 3 (𝑅 ∈ Ring → (MaxIdeal‘𝑅) = {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))})
32eleq2d 2896 . 2 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ 𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))}))
4 neeq1 3068 . . . . 5 (𝑖 = 𝑀 → (𝑖𝐵𝑀𝐵))
5 sseq1 3967 . . . . . . 7 (𝑖 = 𝑀 → (𝑖𝑗𝑀𝑗))
6 eqeq2 2832 . . . . . . . 8 (𝑖 = 𝑀 → (𝑗 = 𝑖𝑗 = 𝑀))
76orbi1d 913 . . . . . . 7 (𝑖 = 𝑀 → ((𝑗 = 𝑖𝑗 = 𝐵) ↔ (𝑗 = 𝑀𝑗 = 𝐵)))
85, 7imbi12d 347 . . . . . 6 (𝑖 = 𝑀 → ((𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)) ↔ (𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
98ralbidv 3184 . . . . 5 (𝑖 = 𝑀 → (∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)) ↔ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
104, 9anbi12d 632 . . . 4 (𝑖 = 𝑀 → ((𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵))) ↔ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
1110elrab 3656 . . 3 (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
12 3anass 1091 . . 3 ((𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ (𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
1311, 12bitr4i 280 . 2 (𝑀 ∈ {𝑖 ∈ (LIdeal‘𝑅) ∣ (𝑖𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = 𝐵)))} ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵))))
143, 13syl6bb 289 1 (𝑅 ∈ Ring → (𝑀 ∈ (MaxIdeal‘𝑅) ↔ (𝑀 ∈ (LIdeal‘𝑅) ∧ 𝑀𝐵 ∧ ∀𝑗 ∈ (LIdeal‘𝑅)(𝑀𝑗 → (𝑗 = 𝑀𝑗 = 𝐵)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wo 843  w3a 1083   = wceq 1537  wcel 2114  wne 3006  wral 3125  {crab 3129  wss 3909  cfv 6327  Basecbs 16458  Ringcrg 19272  LIdealclidl 19914  MaxIdealcmxidl 30975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-iota 6286  df-fun 6329  df-fv 6335  df-mxidl 30976
This theorem is referenced by:  mxidlidl  30979  mxidlnr  30980  mxidlmax  30981  ssmxidl  30986
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