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Theorem lnrfg 30663
Description: Finitely-generated modules over a Noetherian ring, being homomorphic images of free modules, are Noetherian. (Contributed by Stefan O'Rear, 7-Feb-2015.)
Hypothesis
Ref Expression
lnrfg.s  |-  S  =  (Scalar `  M )
Assertion
Ref Expression
lnrfg  |-  ( ( M  e. LFinGen  /\  S  e. LNoeR
)  ->  M  e. LNoeM )

Proof of Theorem lnrfg
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fglmod 30614 . . . . 5  |-  ( M  e. LFinGen  ->  M  e.  LMod )
2 eqid 2462 . . . . . 6  |-  ( Base `  M )  =  (
Base `  M )
3 eqid 2462 . . . . . 6  |-  ( LSpan `  M )  =  (
LSpan `  M )
42, 3islmodfg 30610 . . . . 5  |-  ( M  e.  LMod  ->  ( M  e. LFinGen 
<->  E. a  e.  ~P  ( Base `  M )
( a  e.  Fin  /\  ( ( LSpan `  M
) `  a )  =  ( Base `  M
) ) ) )
51, 4syl 16 . . . 4  |-  ( M  e. LFinGen  ->  ( M  e. LFinGen  <->  E. a  e.  ~P  ( Base `  M ) ( a  e.  Fin  /\  ( ( LSpan `  M
) `  a )  =  ( Base `  M
) ) ) )
65ibi 241 . . 3  |-  ( M  e. LFinGen  ->  E. a  e.  ~P  ( Base `  M )
( a  e.  Fin  /\  ( ( LSpan `  M
) `  a )  =  ( Base `  M
) ) )
76adantr 465 . 2  |-  ( ( M  e. LFinGen  /\  S  e. LNoeR
)  ->  E. a  e.  ~P  ( Base `  M
) ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )
8 eqid 2462 . . . . . 6  |-  ( S freeLMod  a )  =  ( S freeLMod  a )
9 eqid 2462 . . . . . 6  |-  ( Base `  ( S freeLMod  a )
)  =  ( Base `  ( S freeLMod  a )
)
10 eqid 2462 . . . . . 6  |-  ( .s
`  M )  =  ( .s `  M
)
11 eqid 2462 . . . . . 6  |-  ( b  e.  ( Base `  ( S freeLMod  a ) )  |->  ( M  gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )  =  ( b  e.  ( Base `  ( S freeLMod  a ) )  |->  ( M  gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )
121ad3antrrr 729 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  M  e.  LMod )
13 vex 3111 . . . . . . 7  |-  a  e. 
_V
1413a1i 11 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  a  e.  _V )
15 lnrfg.s . . . . . . 7  |-  S  =  (Scalar `  M )
1615a1i 11 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  S  =  (Scalar `  M )
)
17 f1oi 5844 . . . . . . . . 9  |-  (  _I  |`  a ) : a -1-1-onto-> a
18 f1of 5809 . . . . . . . . 9  |-  ( (  _I  |`  a ) : a -1-1-onto-> a  ->  (  _I  |`  a ) : a --> a )
1917, 18ax-mp 5 . . . . . . . 8  |-  (  _I  |`  a ) : a --> a
20 elpwi 4014 . . . . . . . 8  |-  ( a  e.  ~P ( Base `  M )  ->  a  C_  ( Base `  M
) )
21 fss 5732 . . . . . . . 8  |-  ( ( (  _I  |`  a
) : a --> a  /\  a  C_  ( Base `  M ) )  ->  (  _I  |`  a
) : a --> (
Base `  M )
)
2219, 20, 21sylancr 663 . . . . . . 7  |-  ( a  e.  ~P ( Base `  M )  ->  (  _I  |`  a ) : a --> ( Base `  M
) )
2322ad2antlr 726 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  (  _I  |`  a ) : a --> ( Base `  M
) )
248, 9, 2, 10, 11, 12, 14, 16, 23frlmup1 18594 . . . . 5  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  (
b  e.  ( Base `  ( S freeLMod  a )
)  |->  ( M  gsumg  ( b  oF ( .s
`  M ) (  _I  |`  a )
) ) )  e.  ( ( S freeLMod  a
) LMHom  M ) )
25 simpllr 758 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  S  e. LNoeR )
26 simprl 755 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  a  e.  Fin )
278lnrfrlm 30662 . . . . . 6  |-  ( ( S  e. LNoeR  /\  a  e.  Fin )  ->  ( S freeLMod  a )  e. LNoeM )
2825, 26, 27syl2anc 661 . . . . 5  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  ( S freeLMod  a )  e. LNoeM )
298, 9, 2, 10, 11, 12, 14, 16, 23, 3frlmup3 18596 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  ran  ( b  e.  (
Base `  ( S freeLMod  a ) )  |->  ( M 
gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )  =  ( ( LSpan `  M ) `  ran  (  _I  |`  a ) ) )
30 rnresi 5343 . . . . . . . 8  |-  ran  (  _I  |`  a )  =  a
3130fveq2i 5862 . . . . . . 7  |-  ( (
LSpan `  M ) `  ran  (  _I  |`  a
) )  =  ( ( LSpan `  M ) `  a )
32 simprr 756 . . . . . . 7  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  (
( LSpan `  M ) `  a )  =  (
Base `  M )
)
3331, 32syl5eq 2515 . . . . . 6  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  (
( LSpan `  M ) `  ran  (  _I  |`  a
) )  =  (
Base `  M )
)
3429, 33eqtrd 2503 . . . . 5  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  ran  ( b  e.  (
Base `  ( S freeLMod  a ) )  |->  ( M 
gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )  =  ( Base `  M
) )
352lnmepi 30626 . . . . 5  |-  ( ( ( b  e.  (
Base `  ( S freeLMod  a ) )  |->  ( M 
gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )  e.  ( ( S freeLMod  a ) LMHom  M )  /\  ( S freeLMod  a )  e. LNoeM  /\  ran  ( b  e.  ( Base `  ( S freeLMod  a ) )  |->  ( M  gsumg  ( b  oF ( .s `  M
) (  _I  |`  a
) ) ) )  =  ( Base `  M
) )  ->  M  e. LNoeM )
3624, 28, 34, 35syl3anc 1223 . . . 4  |-  ( ( ( ( M  e. LFinGen  /\  S  e. LNoeR )  /\  a  e.  ~P ( Base `  M ) )  /\  ( a  e. 
Fin  /\  ( ( LSpan `  M ) `  a )  =  (
Base `  M )
) )  ->  M  e. LNoeM )
3736exp31 604 . . 3  |-  ( ( M  e. LFinGen  /\  S  e. LNoeR
)  ->  ( a  e.  ~P ( Base `  M
)  ->  ( (
a  e.  Fin  /\  ( ( LSpan `  M
) `  a )  =  ( Base `  M
) )  ->  M  e. LNoeM ) ) )
3837rexlimdv 2948 . 2  |-  ( ( M  e. LFinGen  /\  S  e. LNoeR
)  ->  ( E. a  e.  ~P  ( Base `  M ) ( a  e.  Fin  /\  ( ( LSpan `  M
) `  a )  =  ( Base `  M
) )  ->  M  e. LNoeM ) )
397, 38mpd 15 1  |-  ( ( M  e. LFinGen  /\  S  e. LNoeR
)  ->  M  e. LNoeM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2810   _Vcvv 3108    C_ wss 3471   ~Pcpw 4005    |-> cmpt 4500    _I cid 4785   ran crn 4995    |` cres 4996   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581  (class class class)co 6277    oFcof 6515   Fincfn 7508   Basecbs 14481  Scalarcsca 14549   .scvsca 14550    gsumg cgsu 14687   LModclmod 17290   LSpanclspn 17395   LMHom clmhm 17443   freeLMod cfrlm 18539  LFinGenclfig 30608  LNoeMclnm 30616  LNoeRclnr 30653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-inf2 8049  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-iin 4323  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-se 4834  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-supp 6894  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7462  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-fsupp 7821  df-sup 7892  df-oi 7926  df-card 8311  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-10 10593  df-n0 10787  df-z 10856  df-dec 10968  df-uz 11074  df-fz 11664  df-fzo 11784  df-seq 12066  df-hash 12363  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-sets 14487  df-ress 14488  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-ip 14564  df-tset 14565  df-ple 14566  df-ds 14568  df-hom 14570  df-cco 14571  df-0g 14688  df-gsum 14689  df-prds 14694  df-pws 14696  df-mre 14832  df-mrc 14833  df-acs 14835  df-mnd 15723  df-mhm 15772  df-submnd 15773  df-grp 15853  df-minusg 15854  df-sbg 15855  df-mulg 15856  df-subg 15988  df-ghm 16055  df-cntz 16145  df-lsm 16447  df-cmn 16591  df-abl 16592  df-mgp 16927  df-ur 16939  df-rng 16983  df-subrg 17205  df-lmod 17292  df-lss 17357  df-lsp 17396  df-lmhm 17446  df-lmim 17447  df-lmic 17448  df-lbs 17499  df-sra 17596  df-rgmod 17597  df-nzr 17683  df-dsmm 18525  df-frlm 18540  df-uvc 18576  df-lfig 30609  df-lnm 30617  df-lnr 30654
This theorem is referenced by:  lnrfgtr  30664
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