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Theorem lnmepi 27174
Description: Epimorphic images of Noetherian modules are Noetherian. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypothesis
Ref Expression
lnmepi.b  |-  B  =  ( Base `  T
)
Assertion
Ref Expression
lnmepi  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )

Proof of Theorem lnmepi
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 lmhmlmod2 16113 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
213ad2ant1 979 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e.  LMod )
3 eqid 2438 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
4 lnmepi.b . . . . . . . . 9  |-  B  =  ( Base `  T
)
53, 4lmhmf 16115 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> B )
653ad2ant1 979 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  F : ( Base `  S
) --> B )
7 simp3 960 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  ran  F  =  B )
8 dffo2 5660 . . . . . . 7  |-  ( F : ( Base `  S
) -onto-> B  <->  ( F :
( Base `  S ) --> B  /\  ran  F  =  B ) )
96, 7, 8sylanbrc 647 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  F : ( Base `  S
) -onto-> B )
10 eqid 2438 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
114, 10lssss 16018 . . . . . 6  |-  ( a  e.  ( LSubSp `  T
)  ->  a  C_  B )
12 foimacnv 5695 . . . . . 6  |-  ( ( F : ( Base `  S ) -onto-> B  /\  a  C_  B )  -> 
( F " ( `' F " a ) )  =  a )
139, 11, 12syl2an 465 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( F " ( `' F "
a ) )  =  a )
1413oveq2d 6100 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  ( F " ( `' F " a ) ) )  =  ( Ts  a ) )
15 eqid 2438 . . . . 5  |-  ( Ts  ( F " ( `' F " a ) ) )  =  ( Ts  ( F " ( `' F " a ) ) )
16 eqid 2438 . . . . 5  |-  ( Ss  ( `' F " a ) )  =  ( Ss  ( `' F " a ) )
17 eqid 2438 . . . . 5  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
18 simpl2 962 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  S  e. LNoeM )
1917, 10lmhmpreima 16129 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  T )
)  ->  ( `' F " a )  e.  ( LSubSp `  S )
)
20193ad2antl1 1120 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( `' F " a )  e.  ( LSubSp `  S )
)
2117, 16lnmlssfg 27169 . . . . . 6  |-  ( ( S  e. LNoeM  /\  ( `' F " a )  e.  ( LSubSp `  S
) )  ->  ( Ss  ( `' F " a ) )  e. LFinGen )
2218, 20, 21syl2anc 644 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ss  ( `' F " a ) )  e. LFinGen )
23 simpl1 961 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  F  e.  ( S LMHom  T ) )
2415, 16, 17, 22, 20, 23lmhmfgima 27173 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  ( F " ( `' F " a ) ) )  e. LFinGen )
2514, 24eqeltrrd 2513 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  a
)  e. LFinGen )
2625ralrimiva 2791 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  A. a  e.  ( LSubSp `  T )
( Ts  a )  e. LFinGen )
2710islnm 27166 . 2  |-  ( T  e. LNoeM 
<->  ( T  e.  LMod  /\ 
A. a  e.  (
LSubSp `  T ) ( Ts  a )  e. LFinGen )
)
282, 26, 27sylanbrc 647 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   `'ccnv 4880   ran crn 4882   "cima 4884   -->wf 5453   -onto->wfo 5455   ` cfv 5457  (class class class)co 6084   Basecbs 13474   ↾s cress 13475   LModclmod 15955   LSubSpclss 16013   LMHom clmhm 16100  LFinGenclfig 27156  LNoeMclnm 27164
This theorem is referenced by:  lnmlmic  27177  pwslnmlem1  27185  lnrfg  27314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-sca 13550  df-vsca 13551  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-ghm 15009  df-mgp 15654  df-rng 15668  df-ur 15670  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lmhm 16103  df-lfig 27157  df-lnm 27165
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