Theorem irredcl 20323

Description: An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
irredn0.i	⊢ 𝐼 = (Irred‘𝑅)
irredcl.b	⊢ 𝐵 = (Base‘𝑅)

Assertion

Ref	Expression
irredcl	⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵)

Proof of Theorem irredcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		irredcl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
2		eqid 2726	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2726	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20320	. 2 ⊢ (𝑋 ∈ 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ∈ (Unit‘𝑅) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥(.r‘𝑅)𝑦) = 𝑋 → (𝑥 ∈ (Unit‘𝑅) ∨ 𝑦 ∈ (Unit‘𝑅)))))
6	5	simp1bi 1142	1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ‘cfv 6536  (class class class)co 7404  Basecbs 17150  ._rcmulr 17204  Unitcui 20254  Irredcir 20255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-irred 20258

This theorem is referenced by:  irredrmul  20326  irredneg  20329  prmirred  21356  irredminply  33292