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Theorem lpirlnr 27298
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 16323 . 2  |-  ( R  e. LPIR  ->  R  e.  Ring )
2 eqid 2436 . . . . . . . 8  |-  (LPIdeal `  R
)  =  (LPIdeal `  R
)
3 eqid 2436 . . . . . . . 8  |-  (RSpan `  R )  =  (RSpan `  R )
4 eqid 2436 . . . . . . . 8  |-  ( Base `  R )  =  (
Base `  R )
52, 3, 4islpidl 16317 . . . . . . 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 471 . . . . 5  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. c  e.  ( Base `  R
) a  =  ( (RSpan `  R ) `  { c } ) )
8 snelpwi 4409 . . . . . . . . . 10  |-  ( c  e.  ( Base `  R
)  ->  { c }  e.  ~P ( Base `  R ) )
98adantl 453 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ~P ( Base `  R ) )
10 snfi 7187 . . . . . . . . . 10  |-  { c }  e.  Fin
1110a1i 11 . . . . . . . . 9  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  Fin )
12 elin 3530 . . . . . . . . 9  |-  ( { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )  <->  ( { c }  e.  ~P ( Base `  R
)  /\  { c }  e.  Fin )
)
139, 11, 12sylanbrc 646 . . . . . . . 8  |-  ( ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R
) )  /\  c  e.  ( Base `  R
) )  ->  { c }  e.  ( ~P ( Base `  R
)  i^i  Fin )
)
14 eqid 2436 . . . . . . . 8  |-  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  { c } )
15 fveq2 5728 . . . . . . . . . 10  |-  ( b  =  { c }  ->  ( (RSpan `  R ) `  b
)  =  ( (RSpan `  R ) `  {
c } ) )
1615eqeq2d 2447 . . . . . . . . 9  |-  ( b  =  { c }  ->  ( ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )  <->  ( (RSpan `  R ) `  { c } )  =  ( (RSpan `  R ) `  {
c } ) ) )
1716rspcev 3052 . . . . . . . 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 644 . . . . . . 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 2442 . . . . . . . 8  |-  ( a  =  ( (RSpan `  R ) `  {
c } )  -> 
( a  =  ( (RSpan `  R ) `  b )  <->  ( (RSpan `  R ) `  {
c } )  =  ( (RSpan `  R
) `  b )
) )
2019rexbidv 2726 . . . . . . 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 214 . . . . . 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 2830 . . . . 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 15 . . . 4  |-  ( ( R  e. LPIR  /\  a  e.  (LPIdeal `  R )
)  ->  E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
2423ralrimiva 2789 . . 3  |-  ( R  e. LPIR  ->  A. a  e.  (LPIdeal `  R ) E. b  e.  ( ~P ( Base `  R )  i^i  Fin ) a  =  ( (RSpan `  R ) `  b ) )
25 eqid 2436 . . . . . 6  |-  (LIdeal `  R )  =  (LIdeal `  R )
262, 25islpir 16320 . . . . 5  |-  ( R  e. LPIR 
<->  ( R  e.  Ring  /\  (LIdeal `  R )  =  (LPIdeal `  R )
) )
2726simprbi 451 . . . 4  |-  ( R  e. LPIR  ->  (LIdeal `  R )  =  (LPIdeal `  R )
)
2827raleqdv 2910 . . 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 224 . 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 27295 . 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 646 1  |-  ( R  e. LPIR  ->  R  e. LNoeR )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706    i^i cin 3319   ~Pcpw 3799   {csn 3814   ` cfv 5454   Fincfn 7109   Basecbs 13469   Ringcrg 15660  LIdealclidl 16242  RSpancrsp 16243  LPIdealclpidl 16312  LPIRclpir 16313  LNoeRclnr 27290
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-sra 16244  df-rgmod 16245  df-lidl 16246  df-rsp 16247  df-lpidl 16314  df-lpir 16315  df-lfig 27143  df-lnm 27151  df-lnr 27291
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