Theorem islpir 19249

Description: Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)

Hypotheses

Ref	Expression
lpival.p	|- P = (LPIdeal ` R )
lpiss.u	|- U = (LIdeal ` R )

Assertion

Ref	Expression
islpir	|- ( R e. LPIR <-> ( R e. Ring /\ U = P ) )

Proof of Theorem islpir

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( r = R -> (LIdeal ` r ) = (LIdeal ` R ) )
2		fveq2 6191	. . . 4 |- ( r = R -> (LPIdeal ` r ) = (LPIdeal ` R ) )
3	1, 2	eqeq12d 2637	. . 3 |- ( r = R -> ( (LIdeal ` r ) = (LPIdeal ` r ) <-> (LIdeal ` R ) = (LPIdeal ` R ) ) )
4		lpiss.u	. . . 4 |- U = (LIdeal ` R )
5		lpival.p	. . . 4 |- P = (LPIdeal ` R )
6	4, 5	eqeq12i 2636	. . 3 |- ( U = P <-> (LIdeal ` R ) = (LPIdeal ` R ) )
7	3, 6	syl6bbr 278	. 2 |- ( r = R -> ( (LIdeal ` r ) = (LPIdeal ` r ) <-> U = P ) )
8		df-lpir 19244	. 2 |- LPIR = { r e. Ring | (LIdeal ` r ) = (LPIdeal ` r ) }
9	7, 8	elrab2 3366	1 |- ( R e. LPIR <-> ( R e. Ring /\ U = P ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888   Ringcrg 18547  LIdealclidl 19170  LPIdealclpidl 19241  LPIRclpir 19242

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896  df-lpir 19244

This theorem is referenced by:  islpir2  19251  lpirring  19252  lpirlnr  37687