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Theorem islmodfg 27270
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 27269 . . . 4  |- LFinGen  =  {
a  e.  LMod  |  (
Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) }
21eleq2i 2360 . . 3  |-  ( W  e. LFinGen 
<->  W  e.  { a  e.  LMod  |  ( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) } )
3 fveq2 5541 . . . . 5  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
4 fveq2 5541 . . . . . . . 8  |-  ( a  =  W  ->  ( LSpan `  a )  =  ( LSpan `  W )
)
5 islmodfg.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
64, 5syl6eqr 2346 . . . . . . 7  |-  ( a  =  W  ->  ( LSpan `  a )  =  N )
76imaeq1d 5027 . . . . . 6  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  a
)  i^i  Fin )
) )
83pweqd 3643 . . . . . . . 8  |-  ( a  =  W  ->  ~P ( Base `  a )  =  ~P ( Base `  W
) )
98ineq1d 3382 . . . . . . 7  |-  ( a  =  W  ->  ( ~P ( Base `  a
)  i^i  Fin )  =  ( ~P ( Base `  W )  i^i 
Fin ) )
109imaeq2d 5028 . . . . . 6  |-  ( a  =  W  ->  ( N " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
117, 10eqtrd 2328 . . . . 5  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
123, 11eleq12d 2364 . . . 4  |-  ( a  =  W  ->  (
( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
)  <->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
1312elrab3 2937 . . 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 )
) ) )
142, 13syl5bb 248 . 2  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
15 eqid 2296 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
16 eqid 2296 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1715, 16, 5lspf 15747 . . . . 5  |-  ( W  e.  LMod  ->  N : ~P ( Base `  W
) --> ( LSubSp `  W
) )
18 ffn 5405 . . . . 5  |-  ( N : ~P ( Base `  W ) --> ( LSubSp `  W )  ->  N  Fn  ~P ( Base `  W
) )
1917, 18syl 15 . . . 4  |-  ( W  e.  LMod  ->  N  Fn  ~P ( Base `  W
) )
20 inss1 3402 . . . 4  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
21 fvelimab 5594 . . . 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 ) ) )
2219, 20, 21sylancl 643 . . 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 ) ) )
23 elin 3371 . . . . . . 7  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P ( Base `  W
)  /\  b  e.  Fin ) )
24 islmodfg.b . . . . . . . . . . 11  |-  B  =  ( Base `  W
)
2524eqcomi 2300 . . . . . . . . . 10  |-  ( Base `  W )  =  B
2625pweqi 3642 . . . . . . . . 9  |-  ~P ( Base `  W )  =  ~P B
2726eleq2i 2360 . . . . . . . 8  |-  ( b  e.  ~P ( Base `  W )  <->  b  e.  ~P B )
2827anbi1i 676 . . . . . . 7  |-  ( ( b  e.  ~P ( Base `  W )  /\  b  e.  Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2923, 28bitri 240 . . . . . 6  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
3025eqeq2i 2306 . . . . . 6  |-  ( ( N `  b )  =  ( Base `  W
)  <->  ( N `  b )  =  B )
3129, 30anbi12i 678 . . . . 5  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( (
b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  B ) )
32 anass 630 . . . . 5  |-  ( ( ( b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  B )  <->  ( b  e. 
~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3331, 32bitri 240 . . . 4  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( b  e.  ~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3433rexbii2 2585 . . 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 ) )
3522, 34syl6bb 252 . 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 ) ) )
3614, 35bitrd 244 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271   Fincfn 6879   Basecbs 13164   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744  LFinGenclfig 27268
This theorem is referenced by:  islssfg  27271  lnrfg  27426
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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lfig 27269
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