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Theorem lnmepi 31006
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 17656 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
213ad2ant1 1018 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e.  LMod )
3 eqid 2443 . . . . . . . . 9  |-  ( Base `  S )  =  (
Base `  S )
4 lnmepi.b . . . . . . . . 9  |-  B  =  ( Base `  T
)
53, 4lmhmf 17658 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> B )
653ad2ant1 1018 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  F : ( Base `  S
) --> B )
7 simp3 999 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  ran  F  =  B )
8 dffo2 5789 . . . . . . 7  |-  ( F : ( Base `  S
) -onto-> B  <->  ( F :
( Base `  S ) --> B  /\  ran  F  =  B ) )
96, 7, 8sylanbrc 664 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  F : ( Base `  S
) -onto-> B )
10 eqid 2443 . . . . . . 7  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
114, 10lssss 17561 . . . . . 6  |-  ( a  e.  ( LSubSp `  T
)  ->  a  C_  B )
12 foimacnv 5823 . . . . . 6  |-  ( ( F : ( Base `  S ) -onto-> B  /\  a  C_  B )  -> 
( F " ( `' F " a ) )  =  a )
139, 11, 12syl2an 477 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( F " ( `' F "
a ) )  =  a )
1413oveq2d 6297 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  ( F " ( `' F " a ) ) )  =  ( Ts  a ) )
15 eqid 2443 . . . . 5  |-  ( Ts  ( F " ( `' F " a ) ) )  =  ( Ts  ( F " ( `' F " a ) ) )
16 eqid 2443 . . . . 5  |-  ( Ss  ( `' F " a ) )  =  ( Ss  ( `' F " a ) )
17 eqid 2443 . . . . 5  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
18 simpl2 1001 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  S  e. LNoeM )
1917, 10lmhmpreima 17672 . . . . . . 7  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  T )
)  ->  ( `' F " a )  e.  ( LSubSp `  S )
)
20193ad2antl1 1159 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( `' F " a )  e.  ( LSubSp `  S )
)
2117, 16lnmlssfg 31001 . . . . . 6  |-  ( ( S  e. LNoeM  /\  ( `' F " a )  e.  ( LSubSp `  S
) )  ->  ( Ss  ( `' F " a ) )  e. LFinGen )
2218, 20, 21syl2anc 661 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ss  ( `' F " a ) )  e. LFinGen )
23 simpl1 1000 . . . . 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 31005 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  ( F " ( `' F " a ) ) )  e. LFinGen )
2514, 24eqeltrrd 2532 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  S  e. LNoeM  /\  ran  F  =  B )  /\  a  e.  ( LSubSp `  T )
)  ->  ( Ts  a
)  e. LFinGen )
2625ralrimiva 2857 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  A. a  e.  ( LSubSp `  T )
( Ts  a )  e. LFinGen )
2710islnm 30998 . 2  |-  ( T  e. LNoeM 
<->  ( T  e.  LMod  /\ 
A. a  e.  (
LSubSp `  T ) ( Ts  a )  e. LFinGen )
)
282, 26, 27sylanbrc 664 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  S  e. LNoeM  /\  ran  F  =  B )  ->  T  e. LNoeM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793    C_ wss 3461   `'ccnv 4988   ran crn 4990   "cima 4992   -->wf 5574   -onto->wfo 5576   ` cfv 5578  (class class class)co 6281   Basecbs 14613   ↾s cress 14614   LModclmod 17490   LSubSpclss 17556   LMHom clmhm 17643  LFinGenclfig 30988  LNoeMclnm 30996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-ndx 14616  df-slot 14617  df-base 14618  df-sets 14619  df-ress 14620  df-plusg 14691  df-sca 14694  df-vsca 14695  df-0g 14820  df-mgm 15850  df-sgrp 15889  df-mnd 15899  df-grp 16035  df-minusg 16036  df-sbg 16037  df-subg 16176  df-ghm 16243  df-mgp 17120  df-ur 17132  df-ring 17178  df-lmod 17492  df-lss 17557  df-lsp 17596  df-lmhm 17646  df-lfig 30989  df-lnm 30997
This theorem is referenced by:  lnmlmic  31009  pwslnmlem1  31013  lnrfg  31043
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