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Theorem ssmxidl 33482
Description: Let 𝑅 be a ring, and let 𝐼 be a proper ideal of 𝑅. Then there is a maximal ideal of 𝑅 containing 𝐼. (Contributed by Thierry Arnoux, 10-Apr-2024.)
Hypothesis
Ref Expression
ssmxidl.1 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
ssmxidl ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)
Distinct variable groups:   𝐵,𝑚   𝑚,𝐼   𝑅,𝑚

Proof of Theorem ssmxidl
Dummy variables 𝑗 𝑝 𝑧 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neeq1 3001 . . . . . 6 (𝑝 = 𝐼 → (𝑝𝐵𝐼𝐵))
2 sseq2 4022 . . . . . 6 (𝑝 = 𝐼 → (𝐼𝑝𝐼𝐼))
31, 2anbi12d 632 . . . . 5 (𝑝 = 𝐼 → ((𝑝𝐵𝐼𝑝) ↔ (𝐼𝐵𝐼𝐼)))
4 simp2 1136 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼 ∈ (LIdeal‘𝑅))
5 simp3 1137 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼𝐵)
6 ssidd 4019 . . . . . 6 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼𝐼)
75, 6jca 511 . . . . 5 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → (𝐼𝐵𝐼𝐼))
83, 4, 7elrabd 3697 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → 𝐼 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
98ne0d 4348 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ≠ ∅)
10 ssmxidl.1 . . . . . 6 𝐵 = (Base‘𝑅)
11 eqid 2735 . . . . . 6 {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} = {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}
12 simpl1 1190 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑅 ∈ Ring)
13 simpl2 1191 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝐼 ∈ (LIdeal‘𝑅))
14 simpl3 1192 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝐼𝐵)
15 simpr1 1193 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
16 simpr2 1194 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ≠ ∅)
17 simpr3 1195 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → [] Or 𝑧)
1810, 11, 12, 13, 14, 15, 16, 17ssmxidllem 33481 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧)) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
1918ex 412 . . . 4 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}))
2019alrimiv 1925 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∀𝑧((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}))
21 fvex 6920 . . . . 5 (LIdeal‘𝑅) ∈ V
2221rabex 5345 . . . 4 {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∈ V
2322zornn0 10546 . . 3 (({𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ≠ ∅ ∧ ∀𝑧((𝑧 ⊆ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ 𝑧 ≠ ∅ ∧ [] Or 𝑧) → 𝑧 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})) → ∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
249, 20, 23syl2anc 584 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
25 neeq1 3001 . . . . . . . 8 (𝑝 = 𝑚 → (𝑝𝐵𝑚𝐵))
26 sseq2 4022 . . . . . . . 8 (𝑝 = 𝑚 → (𝐼𝑝𝐼𝑚))
2725, 26anbi12d 632 . . . . . . 7 (𝑝 = 𝑚 → ((𝑝𝐵𝐼𝑝) ↔ (𝑚𝐵𝐼𝑚)))
2827elrab 3695 . . . . . 6 (𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ↔ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
2928anbi2i 623 . . . . 5 (((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ 𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}) ↔ ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))))
30 simpll1 1211 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑅 ∈ Ring)
31 simplrl 777 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚 ∈ (LIdeal‘𝑅))
32 simplr 769 . . . . . . . 8 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
3332simprld 772 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚𝐵)
34 psseq2 4101 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑘 → (𝑚𝑗𝑚𝑘))
3534notbid 318 . . . . . . . . . . . . . . 15 (𝑗 = 𝑘 → (¬ 𝑚𝑗 ↔ ¬ 𝑚𝑘))
36 simp-4r 784 . . . . . . . . . . . . . . 15 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗)
37 neeq1 3001 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑘 → (𝑝𝐵𝑘𝐵))
38 sseq2 4022 . . . . . . . . . . . . . . . . 17 (𝑝 = 𝑘 → (𝐼𝑝𝐼𝑘))
3937, 38anbi12d 632 . . . . . . . . . . . . . . . 16 (𝑝 = 𝑘 → ((𝑝𝐵𝐼𝑝) ↔ (𝑘𝐵𝐼𝑘)))
40 simpllr 776 . . . . . . . . . . . . . . . 16 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 ∈ (LIdeal‘𝑅))
41 simpr 484 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ¬ 𝑘 = 𝐵)
4241neqned 2945 . . . . . . . . . . . . . . . . 17 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘𝐵)
43 simp-5r 786 . . . . . . . . . . . . . . . . . . 19 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚)))
4443simprrd 774 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝐼𝑚)
45 simplr 769 . . . . . . . . . . . . . . . . . 18 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑚𝑘)
4644, 45sstrd 4006 . . . . . . . . . . . . . . . . 17 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝐼𝑘)
4742, 46jca 511 . . . . . . . . . . . . . . . 16 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → (𝑘𝐵𝐼𝑘))
4839, 40, 47elrabd 3697 . . . . . . . . . . . . . . 15 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)})
4935, 36, 48rspcdva 3623 . . . . . . . . . . . . . 14 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → ¬ 𝑚𝑘)
50 npss 4123 . . . . . . . . . . . . . . 15 𝑚𝑘 ↔ (𝑚𝑘𝑚 = 𝑘))
5150biimpi 216 . . . . . . . . . . . . . 14 𝑚𝑘 → (𝑚𝑘𝑚 = 𝑘))
5249, 45, 51sylc 65 . . . . . . . . . . . . 13 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑚 = 𝑘)
5352equcomd 2016 . . . . . . . . . . . 12 (((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) ∧ ¬ 𝑘 = 𝐵) → 𝑘 = 𝑚)
5453ex 412 . . . . . . . . . . 11 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (¬ 𝑘 = 𝐵𝑘 = 𝑚))
5554orrd 863 . . . . . . . . . 10 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (𝑘 = 𝐵𝑘 = 𝑚))
5655orcomd 871 . . . . . . . . 9 ((((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) ∧ 𝑚𝑘) → (𝑘 = 𝑚𝑘 = 𝐵))
5756ex 412 . . . . . . . 8 (((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) ∧ 𝑘 ∈ (LIdeal‘𝑅)) → (𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))
5857ralrimiva 3144 . . . . . . 7 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))
5910ismxidl 33470 . . . . . . . 8 (𝑅 ∈ Ring → (𝑚 ∈ (MaxIdeal‘𝑅) ↔ (𝑚 ∈ (LIdeal‘𝑅) ∧ 𝑚𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))))
6059biimpar 477 . . . . . . 7 ((𝑅 ∈ Ring ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ 𝑚𝐵 ∧ ∀𝑘 ∈ (LIdeal‘𝑅)(𝑚𝑘 → (𝑘 = 𝑚𝑘 = 𝐵)))) → 𝑚 ∈ (MaxIdeal‘𝑅))
6130, 31, 33, 58, 60syl13anc 1371 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝑚 ∈ (MaxIdeal‘𝑅))
6232simprrd 774 . . . . . 6 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → 𝐼𝑚)
6361, 62jca 511 . . . . 5 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ (𝑚 ∈ (LIdeal‘𝑅) ∧ (𝑚𝐵𝐼𝑚))) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚))
6429, 63sylanb 581 . . . 4 ((((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) ∧ 𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}) ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚))
6564expl 457 . . 3 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ((𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ∧ ∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗) → (𝑚 ∈ (MaxIdeal‘𝑅) ∧ 𝐼𝑚)))
6665reximdv2 3162 . 2 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → (∃𝑚 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)}∀𝑗 ∈ {𝑝 ∈ (LIdeal‘𝑅) ∣ (𝑝𝐵𝐼𝑝)} ¬ 𝑚𝑗 → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚))
6724, 66mpd 15 1 ((𝑅 ∈ Ring ∧ 𝐼 ∈ (LIdeal‘𝑅) ∧ 𝐼𝐵) → ∃𝑚 ∈ (MaxIdeal‘𝑅)𝐼𝑚)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 847  w3a 1086  wal 1535   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  wss 3963  wpss 3964  c0 4339   cuni 4912   Or wor 5596  cfv 6563   [] crpss 7741  Basecbs 17245  Ringcrg 20251  LIdealclidl 21234  MaxIdealcmxidl 33467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-ac2 10501  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7742  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-dju 9939  df-card 9977  df-ac 10154  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-lidl 21236  df-mxidl 33468
This theorem is referenced by:  drngmxidlr  33486  krull  33487  zarcls1  33830  zarclssn  33834
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