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Theorem lmhmfgima 27140
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 16101 . . . . . 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 2435 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
10 eqid 2435 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
117, 8, 9, 10islssfg2 27127 . . . . 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 3553 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
1514sseli 3336 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  ~P ( Base `  S
) )
1615elpwid 3800 . . . . . . . 8  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  C_  ( Base `  S
) )
17 eqid 2435 . . . . . . . . 9  |-  ( LSpan `  T )  =  (
LSpan `  T )
1810, 9, 17lmhmlsp 16117 . . . . . . . 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 6089 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) ) )
21 lmhmlmod2 16100 . . . . . . . . 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 5208 . . . . . . . . 9  |-  ( F
" x )  C_  ran  F
25 eqid 2435 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
2610, 25lmhmf 16102 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
273, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( Base `  S ) --> ( Base `  T ) )
28 frn 5589 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
2927, 28syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Base `  T ) )
3024, 29syl5ss 3351 . . . . . . . 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 3554 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
3332sseli 3336 . . . . . . . . 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 5585 . . . . . . . . . . 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 5587 . . . . . . . . . . . 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 3377 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  dom  F )
43 fores 5654 . . . . . . . . 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 7384 . . . . . . . 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 2435 . . . . . . . 8  |-  ( Ts  ( ( LSpan `  T ) `  ( F " x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )
4817, 25, 47islssfgi 27128 . . . . . . 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 2509 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  e. LFinGen )
51 imaeq2 5191 . . . . . . 7  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( F "
( ( LSpan `  S
) `  x )
)  =  ( F
" A ) )
5251oveq2d 6089 . . . . . 6  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( F " A ) ) )
5352eleq1d 2501 . . . . 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 2822 . . 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 2519 1  |-  ( ph  ->  Y  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   dom cdm 4870   ran crn 4871    |` cres 4872   "cima 4873   Fun wfun 5440   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   ↾s cress 13462   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   LMHom clmhm 16087  LFinGenclfig 27123
This theorem is referenced by:  lnmepi  27141
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-ghm 14996  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lmhm 16090  df-lfig 27124
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