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Theorem lpirlnr 30994
Description: Left principal ideal rings are left Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
lpirlnr  |-  ( R  e. LPIR  ->  R  e. LNoeR )

Proof of Theorem lpirlnr
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpirring 17770 . 2  |-  ( R  e. LPIR  ->  R  e.  Ring )
2 eqid 2467 . . . . . . . 8  |-  (LPIdeal `  R
)  =  (LPIdeal `  R
)
3 eqid 2467 . . . . . . . 8  |-  (RSpan `  R )  =  (RSpan `  R )
4 eqid 2467 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
52, 3, 4islpidl 17764 . . . . . . 7  |-  ( R  e.  Ring  ->  ( a  e.  (LPIdeal `  R
)  <->  E. c  e.  (
Base `  R )
a  =  ( (RSpan `  R ) `  {
c } ) ) )
61, 5syl 16 . . . . . 6  |-  ( R  e. LPIR  ->  ( a  e.  (LPIdeal `  R )  <->  E. c  e.  ( Base `  R ) a  =  ( (RSpan `  R
) `  { c } ) ) )
76biimpa 484 . . . . 5  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. c  e.  ( Base `  R
) a  =  ( (RSpan `  R ) `  { c } ) )
8 snelpwi 4698 . . . . . . . . . 10  |-  ( c  e.  ( Base `  R
)  ->  { c }  e.  ~P ( Base `  R ) )
98adantl 466 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ~P ( Base `  R ) )
10 snfi 7608 . . . . . . . . . 10  |-  { c }  e.  Fin
1110a1i 11 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  Fin )
129, 11elind 3693 . . . . . . . 8  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )
)
13 eqid 2467 . . . . . . . 8  |-  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  { c } )
14 fveq2 5872 . . . . . . . . . 10  |-  ( b  =  { c }  ->  ( (RSpan `  R ) `  b
)  =  ( (RSpan `  R ) `  {
c } ) )
1514eqeq2d 2481 . . . . . . . . 9  |-  ( b  =  { c }  ->  ( ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )  <->  ( (RSpan `  R ) `  { c } )  =  ( (RSpan `  R ) `  {
c } ) ) )
1615rspcev 3219 . . . . . . . 8  |-  ( ( { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )  /\  ( (RSpan `  R
) `  { c } )  =  ( (RSpan `  R ) `  { c } ) )  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
)
1712, 13, 16sylancl 662 . . . . . . 7  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
)
18 eqeq1 2471 . . . . . . . 8  |-  ( a  =  ( (RSpan `  R ) `  {
c } )  -> 
( a  =  ( (RSpan `  R ) `  b )  <->  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
) )
1918rexbidv 2978 . . . . . . 7  |-  ( a  =  ( (RSpan `  R ) `  {
c } )  -> 
( E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b )  <->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
) )
2017, 19syl5ibrcom 222 . . . . . 6  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  (
a  =  ( (RSpan `  R ) `  {
c } )  ->  E. b  e.  ( ~P ( Base `  R
)  i^i  Fin )
a  =  ( (RSpan `  R ) `  b
) ) )
2120rexlimdva 2959 . . . . 5  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  ( E. c  e.  ( Base `  R ) a  =  ( (RSpan `  R
) `  { c } )  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) ) )
227, 21mpd 15 . . . 4  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
2322ralrimiva 2881 . . 3  |-  ( R  e. LPIR  ->  A. a  e.  (LPIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
24 eqid 2467 . . . . . 6  |-  (LIdeal `  R )  =  (LIdeal `  R )
252, 24islpir 17767 . . . . 5  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  (LIdeal `  R )  =  (LPIdeal `  R )
) )
2625simprbi 464 . . . 4  |-  ( R  e. LPIR  ->  (LIdeal `  R )  =  (LPIdeal `  R )
)
2726raleqdv 3069 . . 3  |-  ( R  e. LPIR  ->  ( A. a  e.  (LIdeal `  R ) E. b  e.  ( ~P ( Base `  R
)  i^i  Fin )
a  =  ( (RSpan `  R ) `  b
)  <->  A. a  e.  (LPIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) ) )
2823, 27mpbird 232 . 2  |-  ( R  e. LPIR  ->  A. a  e.  (LIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
294, 24, 3islnr2 30991 . 2  |-  ( R  e. LNoeR 
<->  ( R  e.  Ring  /\ 
A. a  e.  (LIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) ) )
301, 28, 29sylanbrc 664 1  |-  ( R  e. LPIR  ->  R  e. LNoeR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    i^i cin 3480   ~Pcpw 4016   {csn 4033   ` cfv 5594   Fincfn 7528   Basecbs 14507   Ringcrg 17070  LIdealclidl 17687  RSpancrsp 17688  LPIdealclpidl 17759  LPIRclpir 17760  LNoeRclnr 30986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-sca 14588  df-vsca 14589  df-ip 14590  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-mgp 17014  df-ur 17026  df-ring 17072  df-subrg 17298  df-lmod 17385  df-lss 17450  df-lsp 17489  df-sra 17689  df-rgmod 17690  df-lidl 17691  df-rsp 17692  df-lpidl 17761  df-lpir 17762  df-lfig 30942  df-lnm 30950  df-lnr 30987
This theorem is referenced by: (None)
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