Theorem irredcl 20370

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 2728	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2728	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20367	. 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 845   = wceq 1533   ∈ wcel 2098  ∀wral 3058  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  ._rcmulr 17241  Unitcui 20301  Irredcir 20302

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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-irred 20305

This theorem is referenced by:  irredrmul  20373  irredneg  20376  prmirred  21407  irrednzr  32968  rprmirredb  33271  irredminply  33417