Theorem irredcl 19861

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 2738	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2738	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 19858	. 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 1143	1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 843   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  ._rcmulr 16889  Unitcui 19796  Irredcir 19797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-irred 19800

This theorem is referenced by:  irredrmul  19864  irredneg  19867  prmirred  20608