Theorem irredcl 20372

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 2737	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
3		irredn0.i	. . 3 ⊢ 𝐼 = (Irred‘𝑅)
4		eqid 2737	. . 3 ⊢ (.r‘𝑅) = (.r‘𝑅)
5	1, 2, 3, 4	isirred2 20369	. 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 1146	1 ⊢ (𝑋 ∈ 𝐼 → 𝑋 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  ._rcmulr 17190  Unitcui 20303  Irredcir 20304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-irred 20307

This theorem is referenced by:  irredrmul  20375  irredneg  20378  prmirred  21441  irrednzr  33344  rprmirredb  33625  irredminply  33894