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Theorem lmhmfgima 35648
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 18191 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
53, 4syl 17 . . . . 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 2429 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
10 eqid 2429 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
117, 8, 9, 10islssfg2 35635 . . . . 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 665 . . . 4  |-  ( ph  ->  ( X  e. LFinGen  <->  E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A ) )
132, 12mpbid 213 . . 3  |-  ( ph  ->  E. x  e.  ( ~P ( Base `  S
)  i^i  Fin )
( ( LSpan `  S
) `  x )  =  A )
14 inss1 3688 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
1514sseli 3466 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  ~P ( Base `  S
) )
1615elpwid 3995 . . . . . . . 8  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  C_  ( Base `  S
) )
17 eqid 2429 . . . . . . . . 9  |-  ( LSpan `  T )  =  (
LSpan `  T )
1810, 9, 17lmhmlsp 18207 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  x
) )  =  ( ( LSpan `  T ) `  ( F " x
) ) )
193, 16, 18syl2an 479 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " ( ( LSpan `  S
) `  x )
)  =  ( (
LSpan `  T ) `  ( F " x ) ) )
2019oveq2d 6321 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) ) )
21 lmhmlmod2 18190 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
223, 21syl 17 . . . . . . . 8  |-  ( ph  ->  T  e.  LMod )
2322adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  T  e.  LMod )
24 imassrn 5199 . . . . . . . . 9  |-  ( F
" x )  C_  ran  F
25 eqid 2429 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
2610, 25lmhmf 18192 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
273, 26syl 17 . . . . . . . . . 10  |-  ( ph  ->  F : ( Base `  S ) --> ( Base `  T ) )
28 frn 5752 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
2927, 28syl 17 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Base `  T ) )
3024, 29syl5ss 3481 . . . . . . . 8  |-  ( ph  ->  ( F " x
)  C_  ( Base `  T ) )
3130adantr 466 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  C_  ( Base `  T ) )
32 inss2 3689 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
3332sseli 3466 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  Fin )
3433adantl 467 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  e.  Fin )
35 ffun 5748 . . . . . . . . . . 11  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
3627, 35syl 17 . . . . . . . . . 10  |-  ( ph  ->  Fun  F )
3736adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  Fun  F )
3816adantl 467 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  ( Base `  S ) )
39 fdm 5750 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
4027, 39syl 17 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  (
Base `  S )
)
4140adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  dom  F  =  ( Base `  S
) )
4238, 41sseqtr4d 3507 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  dom  F )
43 fores 5819 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
4437, 42, 43syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F  |`  x ) : x
-onto-> ( F " x
) )
45 fofi 7866 . . . . . . . 8  |-  ( ( x  e.  Fin  /\  ( F  |`  x ) : x -onto-> ( F
" x ) )  ->  ( F "
x )  e.  Fin )
4634, 44, 45syl2anc 665 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  e.  Fin )
47 eqid 2429 . . . . . . . 8  |-  ( Ts  ( ( LSpan `  T ) `  ( F " x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )
4817, 25, 47islssfgi 35636 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " x )  C_  ( Base `  T )  /\  ( F " x
)  e.  Fin )  ->  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )  e. LFinGen )
4923, 31, 46, 48syl3anc 1264 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  (
( LSpan `  T ) `  ( F " x
) ) )  e. LFinGen )
5020, 49eqeltrd 2517 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  e. LFinGen )
51 imaeq2 5184 . . . . . . 7  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( F "
( ( LSpan `  S
) `  x )
)  =  ( F
" A ) )
5251oveq2d 6321 . . . . . 6  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( F " A ) ) )
5352eleq1d 2498 . . . . 5  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( ( Ts  ( F " ( (
LSpan `  S ) `  x ) ) )  e. LFinGen 
<->  ( Ts  ( F " A ) )  e. LFinGen ) )
5450, 53syl5ibcom 223 . . . 4  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( (
( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" A ) )  e. LFinGen ) )
5554rexlimdva 2924 . . 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 2521 1  |-  ( ph  ->  Y  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783    i^i cin 3441    C_ wss 3442   ~Pcpw 3985   dom cdm 4854   ran crn 4855    |` cres 4856   "cima 4857   Fun wfun 5595   -->wf 5597   -onto->wfo 5599   ` cfv 5601  (class class class)co 6305   Fincfn 7577   Basecbs 15084   ↾s cress 15085   LModclmod 18026   LSubSpclss 18090   LSpanclspn 18129   LMHom clmhm 18177  LFinGenclfig 35631
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-sca 15168  df-vsca 15169  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-ghm 16832  df-mgp 17659  df-ur 17671  df-ring 17717  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lmhm 18180  df-lfig 35632
This theorem is referenced by:  lnmepi  35649
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