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Theorem lmhmfgima 26851
Description: A homomorphism maps finitely generated submodules to finitely generated submodules. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
lmhmfgima.y  |-  Y  =  ( Ts  ( F " A ) )
lmhmfgima.x  |-  X  =  ( Ss  A )
lmhmfgima.u  |-  U  =  ( LSubSp `  S )
lmhmfgima.xf  |-  ( ph  ->  X  e. LFinGen )
lmhmfgima.a  |-  ( ph  ->  A  e.  U )
lmhmfgima.f  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
Assertion
Ref Expression
lmhmfgima  |-  ( ph  ->  Y  e. LFinGen )

Proof of Theorem lmhmfgima
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lmhmfgima.y . 2  |-  Y  =  ( Ts  ( F " A ) )
2 lmhmfgima.xf . . . 4  |-  ( ph  ->  X  e. LFinGen )
3 lmhmfgima.f . . . . . 6  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
4 lmhmlmod1 16036 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
53, 4syl 16 . . . . 5  |-  ( ph  ->  S  e.  LMod )
6 lmhmfgima.a . . . . 5  |-  ( ph  ->  A  e.  U )
7 lmhmfgima.x . . . . . 6  |-  X  =  ( Ss  A )
8 lmhmfgima.u . . . . . 6  |-  U  =  ( LSubSp `  S )
9 eqid 2387 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
10 eqid 2387 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
117, 8, 9, 10islssfg2 26838 . . . . 5  |-  ( ( S  e.  LMod  /\  A  e.  U )  ->  ( X  e. LFinGen  <->  E. x  e.  ( ~P ( Base `  S
)  i^i  Fin )
( ( LSpan `  S
) `  x )  =  A ) )
125, 6, 11syl2anc 643 . . . 4  |-  ( ph  ->  ( X  e. LFinGen  <->  E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A ) )
132, 12mpbid 202 . . 3  |-  ( ph  ->  E. x  e.  ( ~P ( Base `  S
)  i^i  Fin )
( ( LSpan `  S
) `  x )  =  A )
14 inss1 3504 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
1514sseli 3287 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  ~P ( Base `  S
) )
1615elpwid 3751 . . . . . . . 8  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  C_  ( Base `  S
) )
17 eqid 2387 . . . . . . . . 9  |-  ( LSpan `  T )  =  (
LSpan `  T )
1810, 9, 17lmhmlsp 16052 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  x
) )  =  ( ( LSpan `  T ) `  ( F " x
) ) )
193, 16, 18syl2an 464 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " ( ( LSpan `  S
) `  x )
)  =  ( (
LSpan `  T ) `  ( F " x ) ) )
2019oveq2d 6036 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) ) )
21 lmhmlmod2 16035 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
223, 21syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  LMod )
2322adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  T  e.  LMod )
24 imassrn 5156 . . . . . . . . 9  |-  ( F
" x )  C_  ran  F
25 eqid 2387 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
2610, 25lmhmf 16037 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
273, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( Base `  S ) --> ( Base `  T ) )
28 frn 5537 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
2927, 28syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Base `  T ) )
3024, 29syl5ss 3302 . . . . . . . 8  |-  ( ph  ->  ( F " x
)  C_  ( Base `  T ) )
3130adantr 452 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  C_  ( Base `  T ) )
32 inss2 3505 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
3332sseli 3287 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  Fin )
3433adantl 453 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  e.  Fin )
35 ffun 5533 . . . . . . . . . . 11  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
3627, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  F )
3736adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  Fun  F )
3816adantl 453 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  ( Base `  S ) )
39 fdm 5535 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
4027, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  (
Base `  S )
)
4140adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  dom  F  =  ( Base `  S
) )
4238, 41sseqtr4d 3328 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  dom  F )
43 fores 5602 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
4437, 42, 43syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F  |`  x ) : x
-onto-> ( F " x
) )
45 fofi 7328 . . . . . . . 8  |-  ( ( x  e.  Fin  /\  ( F  |`  x ) : x -onto-> ( F
" x ) )  ->  ( F "
x )  e.  Fin )
4634, 44, 45syl2anc 643 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  e.  Fin )
47 eqid 2387 . . . . . . . 8  |-  ( Ts  ( ( LSpan `  T ) `  ( F " x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )
4817, 25, 47islssfgi 26839 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " x )  C_  ( Base `  T )  /\  ( F " x
)  e.  Fin )  ->  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )  e. LFinGen )
4923, 31, 46, 48syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  (
( LSpan `  T ) `  ( F " x
) ) )  e. LFinGen )
5020, 49eqeltrd 2461 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  e. LFinGen )
51 imaeq2 5139 . . . . . . 7  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( F "
( ( LSpan `  S
) `  x )
)  =  ( F
" A ) )
5251oveq2d 6036 . . . . . 6  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( F " A ) ) )
5352eleq1d 2453 . . . . 5  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( ( Ts  ( F " ( (
LSpan `  S ) `  x ) ) )  e. LFinGen 
<->  ( Ts  ( F " A ) )  e. LFinGen ) )
5450, 53syl5ibcom 212 . . . 4  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( (
( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" A ) )  e. LFinGen ) )
5554rexlimdva 2773 . . 3  |-  ( ph  ->  ( E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A  -> 
( Ts  ( F " A ) )  e. LFinGen ) )
5613, 55mpd 15 . 2  |-  ( ph  ->  ( Ts  ( F " A ) )  e. LFinGen )
571, 56syl5eqel 2471 1  |-  ( ph  ->  Y  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   dom cdm 4818   ran crn 4819    |` cres 4820   "cima 4821   Fun wfun 5388   -->wf 5390   -onto->wfo 5392   ` cfv 5394  (class class class)co 6020   Fincfn 7045   Basecbs 13396   ↾s cress 13397   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974   LMHom clmhm 16022  LFinGenclfig 26834
This theorem is referenced by:  lnmepi  26852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-sca 13472  df-vsca 13473  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-ghm 14931  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lmhm 16025  df-lfig 26835
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