Theorem irredcl 18704

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 e. I -> X e. 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 2622	. . 3 |- (Unit ` R ) = (Unit ` R )
3		irredn0.i	. . 3 |- I = (Irred ` R )
4		eqid 2622	. . 3 |- ( .r ` R ) = ( .r ` R )
5	1, 2, 3, 4	isirred2 18701	. 2 |- ( X e. I <-> ( X e. B /\ -. X e. (Unit ` R ) /\ A. x e. B A. y e. B ( ( x ( .r ` R ) y ) = X -> ( x e. (Unit ` R ) \/ y e. (Unit ` R ) ) ) ) )
6	5	simp1bi 1076	1 |- ( X e. I -> X e. B )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .rcmulr 15942  Unitcui 18639  Irredcir 18640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-irred 18643

This theorem is referenced by:  irredrmul  18707  irredneg  18710  prmirred  19843