pridlidl - Mathbox for Jeff Madsen

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Theorem pridlidl 38064

Description: A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)

**Assertion**
Ref	Expression
pridlidl	⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Proof of Theorem pridlidl Dummy variables 𝑥 𝑦 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		eqid 2736	. . . 4 ⊢ (2^nd ‘𝑅) = (2^nd ‘𝑅)
3		eqid 2736	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	ispridl 38063	. . 3 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1^st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2^nd ‘𝑅)𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
5		3anass 1094	. . 3 ⊢ ((𝑃 ∈ (Idl‘𝑅) ∧ 𝑃 ≠ ran (1^st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2^nd ‘𝑅)𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃 ≠ ran (1^st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2^nd ‘𝑅)𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃)))))
6	4, 5	bitrdi 287	. 2 ⊢ (𝑅 ∈ RingOps → (𝑃 ∈ (PrIdl‘𝑅) ↔ (𝑃 ∈ (Idl‘𝑅) ∧ (𝑃 ≠ ran (1^st ‘𝑅) ∧ ∀𝑎 ∈ (Idl‘𝑅)∀𝑏 ∈ (Idl‘𝑅)(∀𝑥 ∈ 𝑎 ∀𝑦 ∈ 𝑏 (𝑥(2^nd ‘𝑅)𝑦) ∈ 𝑃 → (𝑎 ⊆ 𝑃 ∨ 𝑏 ⊆ 𝑃))))))
7	6	simprbda 498	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdl‘𝑅)) → 𝑃 ∈ (Idl‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ⊆ wss 3931 ran crn 5660 ‘cfv 6536 (class class class)co 7410 1^st c1st 7991 2^nd c2nd 7992 RingOpscrngo 37923 Idlcidl 38036 PrIdlcpridl 38037

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-pridl 38040

This theorem is referenced by: (None)

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