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Theorem lmhmfgima 27285
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 15806 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
53, 4syl 15 . . . . 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 2296 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
10 eqid 2296 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
117, 8, 9, 10islssfg2 27272 . . . . 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 642 . . . 4  |-  ( ph  ->  ( X  e. LFinGen  <->  E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A ) )
132, 12mpbid 201 . . 3  |-  ( ph  ->  E. x  e.  ( ~P ( Base `  S
)  i^i  Fin )
( ( LSpan `  S
) `  x )  =  A )
14 inss1 3402 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
1514sseli 3189 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  ~P ( Base `  S
) )
16 elpwi 3646 . . . . . . . . 9  |-  ( x  e.  ~P ( Base `  S )  ->  x  C_  ( Base `  S
) )
1715, 16syl 15 . . . . . . . 8  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  C_  ( Base `  S
) )
18 eqid 2296 . . . . . . . . 9  |-  ( LSpan `  T )  =  (
LSpan `  T )
1910, 9, 18lmhmlsp 15822 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  x
) )  =  ( ( LSpan `  T ) `  ( F " x
) ) )
203, 17, 19syl2an 463 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " ( ( LSpan `  S
) `  x )
)  =  ( (
LSpan `  T ) `  ( F " x ) ) )
2120oveq2d 5890 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) ) )
22 lmhmlmod2 15805 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
233, 22syl 15 . . . . . . . 8  |-  ( ph  ->  T  e.  LMod )
2423adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  T  e.  LMod )
25 imassrn 5041 . . . . . . . . 9  |-  ( F
" x )  C_  ran  F
26 eqid 2296 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
2710, 26lmhmf 15807 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
283, 27syl 15 . . . . . . . . . 10  |-  ( ph  ->  F : ( Base `  S ) --> ( Base `  T ) )
29 frn 5411 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
3028, 29syl 15 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Base `  T ) )
3125, 30syl5ss 3203 . . . . . . . 8  |-  ( ph  ->  ( F " x
)  C_  ( Base `  T ) )
3231adantr 451 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  C_  ( Base `  T ) )
33 inss2 3403 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
3433sseli 3189 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  Fin )
3534adantl 452 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  e.  Fin )
36 ffun 5407 . . . . . . . . . . 11  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
3728, 36syl 15 . . . . . . . . . 10  |-  ( ph  ->  Fun  F )
3837adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  Fun  F )
3917adantl 452 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  ( Base `  S ) )
40 fdm 5409 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
4128, 40syl 15 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  (
Base `  S )
)
4241adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  dom  F  =  ( Base `  S
) )
4339, 42sseqtr4d 3228 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  dom  F )
44 fores 5476 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
4538, 43, 44syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F  |`  x ) : x
-onto-> ( F " x
) )
46 fofi 7158 . . . . . . . 8  |-  ( ( x  e.  Fin  /\  ( F  |`  x ) : x -onto-> ( F
" x ) )  ->  ( F "
x )  e.  Fin )
4735, 45, 46syl2anc 642 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  e.  Fin )
48 eqid 2296 . . . . . . . 8  |-  ( Ts  ( ( LSpan `  T ) `  ( F " x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )
4918, 26, 48islssfgi 27273 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " x )  C_  ( Base `  T )  /\  ( F " x
)  e.  Fin )  ->  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )  e. LFinGen )
5024, 32, 47, 49syl3anc 1182 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  (
( LSpan `  T ) `  ( F " x
) ) )  e. LFinGen )
5121, 50eqeltrd 2370 . . . . 5  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  e. LFinGen )
52 imaeq2 5024 . . . . . . 7  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( F "
( ( LSpan `  S
) `  x )
)  =  ( F
" A ) )
5352oveq2d 5890 . . . . . 6  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( F " A ) ) )
5453eleq1d 2362 . . . . 5  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( ( Ts  ( F " ( (
LSpan `  S ) `  x ) ) )  e. LFinGen 
<->  ( Ts  ( F " A ) )  e. LFinGen ) )
5551, 54syl5ibcom 211 . . . 4  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( (
( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" A ) )  e. LFinGen ) )
5655rexlimdva 2680 . . 3  |-  ( ph  ->  ( E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A  -> 
( Ts  ( F " A ) )  e. LFinGen ) )
5713, 56mpd 14 . 2  |-  ( ph  ->  ( Ts  ( F " A ) )  e. LFinGen )
581, 57syl5eqel 2380 1  |-  ( ph  ->  Y  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   ↾s cress 13165   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LMHom clmhm 15792  LFinGenclfig 27268
This theorem is referenced by:  lnmepi  27286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lmhm 15795  df-lfig 27269
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