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Theorem lmhmfgima 29463
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 17136 . . . . . 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 2443 . . . . . 6  |-  ( LSpan `  S )  =  (
LSpan `  S )
10 eqid 2443 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
117, 8, 9, 10islssfg2 29450 . . . . 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 661 . . . 4  |-  ( ph  ->  ( X  e. LFinGen  <->  E. x  e.  ( ~P ( Base `  S )  i^i  Fin ) ( ( LSpan `  S ) `  x
)  =  A ) )
132, 12mpbid 210 . . 3  |-  ( ph  ->  E. x  e.  ( ~P ( Base `  S
)  i^i  Fin )
( ( LSpan `  S
) `  x )  =  A )
14 inss1 3591 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
1514sseli 3373 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  ~P ( Base `  S
) )
1615elpwid 3891 . . . . . . . 8  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  C_  ( Base `  S
) )
17 eqid 2443 . . . . . . . . 9  |-  ( LSpan `  T )  =  (
LSpan `  T )
1810, 9, 17lmhmlsp 17152 . . . . . . . 8  |-  ( ( F  e.  ( S LMHom 
T )  /\  x  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  x
) )  =  ( ( LSpan `  T ) `  ( F " x
) ) )
193, 16, 18syl2an 477 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " ( ( LSpan `  S
) `  x )
)  =  ( (
LSpan `  T ) `  ( F " x ) ) )
2019oveq2d 6128 . . . . . 6  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( Ts  ( F " ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) ) )
21 lmhmlmod2 17135 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
223, 21syl 16 . . . . . . . 8  |-  ( ph  ->  T  e.  LMod )
2322adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  T  e.  LMod )
24 imassrn 5201 . . . . . . . . 9  |-  ( F
" x )  C_  ran  F
25 eqid 2443 . . . . . . . . . . . 12  |-  ( Base `  T )  =  (
Base `  T )
2610, 25lmhmf 17137 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
273, 26syl 16 . . . . . . . . . 10  |-  ( ph  ->  F : ( Base `  S ) --> ( Base `  T ) )
28 frn 5586 . . . . . . . . . 10  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  ran  F  C_  ( Base `  T )
)
2927, 28syl 16 . . . . . . . . 9  |-  ( ph  ->  ran  F  C_  ( Base `  T ) )
3024, 29syl5ss 3388 . . . . . . . 8  |-  ( ph  ->  ( F " x
)  C_  ( Base `  T ) )
3130adantr 465 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  C_  ( Base `  T ) )
32 inss2 3592 . . . . . . . . . 10  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
3332sseli 3373 . . . . . . . . 9  |-  ( x  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  x  e.  Fin )
3433adantl 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  e.  Fin )
35 ffun 5582 . . . . . . . . . . 11  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  Fun  F )
3627, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  Fun  F )
3736adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  Fun  F )
3816adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  ( Base `  S ) )
39 fdm 5584 . . . . . . . . . . . 12  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  dom  F  =  ( Base `  S
) )
4027, 39syl 16 . . . . . . . . . . 11  |-  ( ph  ->  dom  F  =  (
Base `  S )
)
4140adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  dom  F  =  ( Base `  S
) )
4238, 41sseqtr4d 3414 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  x  C_  dom  F )
43 fores 5650 . . . . . . . . 9  |-  ( ( Fun  F  /\  x  C_ 
dom  F )  -> 
( F  |`  x
) : x -onto-> ( F " x ) )
4437, 42, 43syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F  |`  x ) : x
-onto-> ( F " x
) )
45 fofi 7618 . . . . . . . 8  |-  ( ( x  e.  Fin  /\  ( F  |`  x ) : x -onto-> ( F
" x ) )  ->  ( F "
x )  e.  Fin )
4634, 44, 45syl2anc 661 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( F " x )  e.  Fin )
47 eqid 2443 . . . . . . . 8  |-  ( Ts  ( ( LSpan `  T ) `  ( F " x
) ) )  =  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )
4817, 25, 47islssfgi 29451 . . . . . . 7  |-  ( ( T  e.  LMod  /\  ( F " x )  C_  ( Base `  T )  /\  ( F " x
)  e.  Fin )  ->  ( Ts  ( ( LSpan `  T ) `  ( F " x ) ) )  e. LFinGen )
4923, 31, 46, 48syl3anc 1218 . . . . . 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 5186 . . . . . . 7  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( F "
( ( LSpan `  S
) `  x )
)  =  ( F
" A ) )
5251oveq2d 6128 . . . . . 6  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" ( ( LSpan `  S ) `  x
) ) )  =  ( Ts  ( F " A ) ) )
5352eleq1d 2509 . . . . 5  |-  ( ( ( LSpan `  S ) `  x )  =  A  ->  ( ( Ts  ( F " ( (
LSpan `  S ) `  x ) ) )  e. LFinGen 
<->  ( Ts  ( F " A ) )  e. LFinGen ) )
5450, 53syl5ibcom 220 . . . 4  |-  ( (
ph  /\  x  e.  ( ~P ( Base `  S
)  i^i  Fin )
)  ->  ( (
( LSpan `  S ) `  x )  =  A  ->  ( Ts  ( F
" A ) )  e. LFinGen ) )
5554rexlimdva 2862 . . 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 2527 1  |-  ( ph  ->  Y  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2737    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   dom cdm 4861   ran crn 4862    |` cres 4863   "cima 4864   Fun wfun 5433   -->wf 5435   -onto->wfo 5437   ` cfv 5439  (class class class)co 6112   Fincfn 7331   Basecbs 14195   ↾s cress 14196   LModclmod 16970   LSubSpclss 17035   LSpanclspn 17074   LMHom clmhm 17122  LFinGenclfig 29446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-sca 14275  df-vsca 14276  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-ghm 15766  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lmhm 17125  df-lfig 29447
This theorem is referenced by:  lnmepi  29464
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