Theorem ufdprmidl 33519

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 4179	. . . . 5 ⊢ (𝑗 = 𝐽 → (𝑗 ∩ 𝑃) = (𝐽 ∩ 𝑃))
2	1	neeq1d 2985	. . . 4 ⊢ (𝑗 = 𝐽 → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
3	2	adantl 481	. . 3 ⊢ ((𝜑 ∧ 𝑗 = 𝐽) → ((𝑗 ∩ 𝑃) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅))
4		incom 4175	. . . . 5 ⊢ (𝑃 ∩ 𝐽) = (𝐽 ∩ 𝑃)
5	4	neeq1i 2990	. . . 4 ⊢ ((𝑃 ∩ 𝐽) ≠ ∅ ↔ (𝐽 ∩ 𝑃) ≠ ∅)
6		inn0 4338	. . . 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 4754	. 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 33518	. . . 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 3586	1 ⊢ (𝜑 → ∃𝑝 ∈ 𝑃 𝑝 ∈ 𝐽)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2926  ∀wral 3045  ∃wrex 3054   ∖ cdif 3914   ∩ cin 3916  ∅c0 4299  {csn 4592  ‘cfv 6514  0_gc0g 17409  RPrimecrpm 20348  IDomncidom 20609  PrmIdealcprmidl 33413  UFDcufd 33516

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-11 2158  ax-12 2178  ax-ext 2702

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-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-iota 6467  df-fv 6522  df-ufd 33517

This theorem is referenced by:  1arithufdlem1  33522