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Theorem islmodfg 26836
Description: Property of a finitely generated left module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
islmodfg.b  |-  B  =  ( Base `  W
)
islmodfg.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
islmodfg  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  E. b  e.  ~P  B ( b  e. 
Fin  /\  ( N `  b )  =  B ) ) )
Distinct variable groups:    W, b    B, b    N, b

Proof of Theorem islmodfg
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 df-lfig 26835 . . . 4  |- LFinGen  =  {
a  e.  LMod  |  (
Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) }
21eleq2i 2451 . . 3  |-  ( W  e. LFinGen 
<->  W  e.  { a  e.  LMod  |  ( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) } )
3 fveq2 5668 . . . . 5  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
4 fveq2 5668 . . . . . . 7  |-  ( a  =  W  ->  ( LSpan `  a )  =  ( LSpan `  W )
)
5 islmodfg.n . . . . . . 7  |-  N  =  ( LSpan `  W )
64, 5syl6eqr 2437 . . . . . 6  |-  ( a  =  W  ->  ( LSpan `  a )  =  N )
73pweqd 3747 . . . . . . 7  |-  ( a  =  W  ->  ~P ( Base `  a )  =  ~P ( Base `  W
) )
87ineq1d 3484 . . . . . 6  |-  ( a  =  W  ->  ( ~P ( Base `  a
)  i^i  Fin )  =  ( ~P ( Base `  W )  i^i 
Fin ) )
96, 8imaeq12d 5144 . . . . 5  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
103, 9eleq12d 2455 . . . 4  |-  ( a  =  W  ->  (
( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
)  <->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
1110elrab3 3036 . . 3  |-  ( W  e.  LMod  ->  ( W  e.  { a  e. 
LMod  |  ( Base `  a )  e.  ( ( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) ) }  <-> 
( Base `  W )  e.  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) ) )
122, 11syl5bb 249 . 2  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
13 eqid 2387 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
14 eqid 2387 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 5lspf 15977 . . . . 5  |-  ( W  e.  LMod  ->  N : ~P ( Base `  W
) --> ( LSubSp `  W
) )
16 ffn 5531 . . . . 5  |-  ( N : ~P ( Base `  W ) --> ( LSubSp `  W )  ->  N  Fn  ~P ( Base `  W
) )
1715, 16syl 16 . . . 4  |-  ( W  e.  LMod  ->  N  Fn  ~P ( Base `  W
) )
18 inss1 3504 . . . 4  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
19 fvelimab 5721 . . . 4  |-  ( ( N  Fn  ~P ( Base `  W )  /\  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
) )  ->  (
( Base `  W )  e.  ( N " ( ~P ( Base `  W
)  i^i  Fin )
)  <->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( N `  b
)  =  ( Base `  W ) ) )
2017, 18, 19sylancl 644 . . 3  |-  ( W  e.  LMod  ->  ( (
Base `  W )  e.  ( N " ( ~P ( Base `  W
)  i^i  Fin )
)  <->  E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( N `  b
)  =  ( Base `  W ) ) )
21 elin 3473 . . . . . . 7  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P ( Base `  W
)  /\  b  e.  Fin ) )
22 islmodfg.b . . . . . . . . . . 11  |-  B  =  ( Base `  W
)
2322eqcomi 2391 . . . . . . . . . 10  |-  ( Base `  W )  =  B
2423pweqi 3746 . . . . . . . . 9  |-  ~P ( Base `  W )  =  ~P B
2524eleq2i 2451 . . . . . . . 8  |-  ( b  e.  ~P ( Base `  W )  <->  b  e.  ~P B )
2625anbi1i 677 . . . . . . 7  |-  ( ( b  e.  ~P ( Base `  W )  /\  b  e.  Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2721, 26bitri 241 . . . . . 6  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2823eqeq2i 2397 . . . . . 6  |-  ( ( N `  b )  =  ( Base `  W
)  <->  ( N `  b )  =  B )
2927, 28anbi12i 679 . . . . 5  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( (
b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  B ) )
30 anass 631 . . . . 5  |-  ( ( ( b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  B )  <->  ( b  e. 
~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3129, 30bitri 241 . . . 4  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( b  e.  ~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3231rexbii2 2678 . . 3  |-  ( E. b  e.  ( ~P ( Base `  W
)  i^i  Fin )
( N `  b
)  =  ( Base `  W )  <->  E. b  e.  ~P  B ( b  e.  Fin  /\  ( N `  b )  =  B ) )
3320, 32syl6bb 253 . 2  |-  ( W  e.  LMod  ->  ( (
Base `  W )  e.  ( N " ( ~P ( Base `  W
)  i^i  Fin )
)  <->  E. b  e.  ~P  B ( b  e. 
Fin  /\  ( N `  b )  =  B ) ) )
3412, 33bitrd 245 1  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  E. b  e.  ~P  B ( b  e. 
Fin  /\  ( N `  b )  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2650   {crab 2653    i^i cin 3262    C_ wss 3263   ~Pcpw 3742   "cima 4821    Fn wfn 5389   -->wf 5390   ` cfv 5394   Fincfn 7045   Basecbs 13396   LModclmod 15877   LSubSpclss 15935   LSpanclspn 15974  LFinGenclfig 26834
This theorem is referenced by:  islssfg  26837  lnrfg  26992
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-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  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-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-plusg 13469  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lfig 26835
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