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Theorem lpirlnr 27424
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 lpirrng 16020 . 2  |-  ( R  e. LPIR  ->  R  e.  Ring )
2 eqid 2296 . . . . . . . 8  |-  (LPIdeal `  R
)  =  (LPIdeal `  R
)
3 eqid 2296 . . . . . . . 8  |-  (RSpan `  R )  =  (RSpan `  R )
4 eqid 2296 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
52, 3, 4islpidl 16014 . . . . . . 7  |-  ( R  e.  Ring  ->  ( a  e.  (LPIdeal `  R
)  <->  E. c  e.  (
Base `  R )
a  =  ( (RSpan `  R ) `  {
c } ) ) )
61, 5syl 15 . . . . . 6  |-  ( R  e. LPIR  ->  ( a  e.  (LPIdeal `  R )  <->  E. c  e.  ( Base `  R ) a  =  ( (RSpan `  R
) `  { c } ) ) )
76biimpa 470 . . . . 5  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. c  e.  ( Base `  R
) a  =  ( (RSpan `  R ) `  { c } ) )
8 snelpwi 4236 . . . . . . . . . 10  |-  ( c  e.  ( Base `  R
)  ->  { c }  e.  ~P ( Base `  R ) )
98adantl 452 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ~P ( Base `  R ) )
10 snfi 6957 . . . . . . . . . 10  |-  { c }  e.  Fin
1110a1i 10 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  Fin )
12 elin 3371 . . . . . . . . 9  |-  ( { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )  <->  ( { c }  e.  ~P ( Base `  R
)  /\  { c }  e.  Fin )
)
139, 11, 12sylanbrc 645 . . . . . . . 8  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )
)
14 eqid 2296 . . . . . . . 8  |-  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  { c } )
15 fveq2 5541 . . . . . . . . . 10  |-  ( b  =  { c }  ->  ( (RSpan `  R ) `  b
)  =  ( (RSpan `  R ) `  {
c } ) )
1615eqeq2d 2307 . . . . . . . . 9  |-  ( b  =  { c }  ->  ( ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )  <->  ( (RSpan `  R ) `  { c } )  =  ( (RSpan `  R ) `  {
c } ) ) )
1716rspcev 2897 . . . . . . . 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 )
)
1813, 14, 17sylancl 643 . . . . . . 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 )
)
19 eqeq1 2302 . . . . . . . 8  |-  ( a  =  ( (RSpan `  R ) `  {
c } )  -> 
( a  =  ( (RSpan `  R ) `  b )  <->  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
) )
2019rexbidv 2577 . . . . . . 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 )
) )
2118, 20syl5ibrcom 213 . . . . . 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
) ) )
2221rexlimdva 2680 . . . . 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 ) ) )
237, 22mpd 14 . . . 4  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
2423ralrimiva 2639 . . 3  |-  ( R  e. LPIR  ->  A. a  e.  (LPIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
25 eqid 2296 . . . . . 6  |-  (LIdeal `  R )  =  (LIdeal `  R )
262, 25islpir 16017 . . . . 5  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  (LIdeal `  R )  =  (LPIdeal `  R )
) )
2726simprbi 450 . . . 4  |-  ( R  e. LPIR  ->  (LIdeal `  R )  =  (LPIdeal `  R )
)
2827raleqdv 2755 . . 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 ) ) )
2924, 28mpbird 223 . 2  |-  ( R  e. LPIR  ->  A. a  e.  (LIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
304, 25, 3islnr2 27421 . 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 ) ) )
311, 29, 30sylanbrc 645 1  |-  ( R  e. LPIR  ->  R  e. LNoeR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    i^i cin 3164   ~Pcpw 3638   {csn 3653   ` cfv 5271   Fincfn 6879   Basecbs 13164   Ringcrg 15353  LIdealclidl 15939  RSpancrsp 15940  LPIdealclpidl 16009  LPIRclpir 16010  LNoeRclnr 27416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lsp 15745  df-sra 15941  df-rgmod 15942  df-lidl 15943  df-rsp 15944  df-lpidl 16011  df-lpir 16012  df-lfig 27269  df-lnm 27277  df-lnr 27417
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