Theorem ufdprmidl 33548

Description: In a unique factorization domain 𝑅, a nonzero prime ideal 𝐽 contains a prime element 𝑝. (Contributed by Thierry Arnoux, 3-Jun-2025.)

Hypotheses

Ref	Expression
isufd.i	⊢ 𝐼 = (PrmIdeal‘𝑅)
isufd.3	⊢ 𝑃 = (RPrime‘𝑅)
isufd.0	⊢ 0 = (0g‘𝑅)
ufdprmidl.2	⊢ (𝜑 → 𝑅 ∈ UFD)
ufdprmidl.3	⊢ (𝜑 → 𝐽 ∈ 𝐼)
ufdprmidl.4	⊢ (𝜑 → 𝐽 ≠ { 0 })

Assertion

Ref	Expression
ufdprmidl	⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)

Distinct variable groups:   𝐽,𝑝   𝑃,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝑅(𝑝)   𝐼(𝑝)   0 (𝑝)

Proof of Theorem ufdprmidl

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 4220	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝑃) = (𝐽 ∩ 𝑃))
2	1	neeq1d 2997	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
3	2	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
4		incom 4216	. . . . 5 ⊢ (𝑃 ∩ 𝐽) = (𝐽 ∩ 𝑃)
5	4	neeq1i 3002	. . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅)
6		inn0 4377	. . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
7	5, 6	bitr3i 277	. . 3 ⊢ ((𝐽 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)
8	3, 7	bitrdi 287	. 2 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽))
9		ufdprmidl.3	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐼)
10		ufdprmidl.4	. . 3 ⊢ (𝜑 → 𝐽 ≠ { 0 })
11	9, 10	eldifsnd 4791	. 2 ⊢ (𝜑 → 𝐽 ∈ (𝐼 ∖ {{ 0 }}))
12		ufdprmidl.2	. . 3 ⊢ (𝜑 → 𝑅 ∈ UFD)
13		isufd.i	. . . . 5 ⊢ 𝐼 = (PrmIdeal‘𝑅)
14		isufd.3	. . . . 5 ⊢ 𝑃 = (RPrime‘𝑅)
15		isufd.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
16	13, 14, 15	isufd 33547	. . . 4 ⊢ (𝑅 ∈ UFD ↔ (𝑅 ∈ IDomn ∧ ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅))
17	16	simprbi 496	. . 3 ⊢ (𝑅 ∈ UFD → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
18	12, 17	syl 17	. 2 ⊢ (𝜑 → ∀𝑗 ∈ (𝐼 ∖ {{ 0 }})(𝑗 ∩ 𝑃) ≠ ∅)
19	8, 11, 18	rspcdv2 3616	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067   ∖ cdif 3959   ∩ cin 3961  ∅c0 4338  {csn 4630  ‘cfv 6562  0_gc0g 17485  RPrimecrpm 20448  IDomncidom 20709  PrmIdealcprmidl 33442  UFDcufd 33545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ufd 33546

This theorem is referenced by:  1arithufdlem1  33551