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Theorem islmodfg 30647
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 30646 . . . 4  |- LFinGen  =  {
a  e.  LMod  |  (
Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) }
21eleq2i 2545 . . 3  |-  ( W  e. LFinGen 
<->  W  e.  { a  e.  LMod  |  ( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) } )
3 fveq2 5866 . . . . 5  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
4 fveq2 5866 . . . . . . 7  |-  ( a  =  W  ->  ( LSpan `  a )  =  ( LSpan `  W )
)
5 islmodfg.n . . . . . . 7  |-  N  =  ( LSpan `  W )
64, 5syl6eqr 2526 . . . . . 6  |-  ( a  =  W  ->  ( LSpan `  a )  =  N )
73pweqd 4015 . . . . . . 7  |-  ( a  =  W  ->  ~P ( Base `  a )  =  ~P ( Base `  W
) )
87ineq1d 3699 . . . . . 6  |-  ( a  =  W  ->  ( ~P ( Base `  a
)  i^i  Fin )  =  ( ~P ( Base `  W )  i^i 
Fin ) )
96, 8imaeq12d 5338 . . . . 5  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
103, 9eleq12d 2549 . . . 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 3262 . . 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 257 . 2  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
13 eqid 2467 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
14 eqid 2467 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 5lspf 17420 . . . . 5  |-  ( W  e.  LMod  ->  N : ~P ( Base `  W
) --> ( LSubSp `  W
) )
16 ffn 5731 . . . . 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 3718 . . . 4  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
19 fvelimab 5923 . . . 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 662 . . 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 3687 . . . . . . 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 2480 . . . . . . . . . 10  |-  ( Base `  W )  =  B
2423pweqi 4014 . . . . . . . . 9  |-  ~P ( Base `  W )  =  ~P B
2524eleq2i 2545 . . . . . . . 8  |-  ( b  e.  ~P ( Base `  W )  <->  b  e.  ~P B )
2625anbi1i 695 . . . . . . 7  |-  ( ( b  e.  ~P ( Base `  W )  /\  b  e.  Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2721, 26bitri 249 . . . . . 6  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2823eqeq2i 2485 . . . . . 6  |-  ( ( N `  b )  =  ( Base `  W
)  <->  ( N `  b )  =  B )
2927, 28anbi12i 697 . . . . 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 649 . . . . 5  |-  ( ( ( b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  B )  <->  ( b  e. 
~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3129, 30bitri 249 . . . 4  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( b  e.  ~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3231rexbii2 2963 . . 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 261 . 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 253 1  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  E. b  e.  ~P  B ( b  e. 
Fin  /\  ( N `  b )  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588   Fincfn 7516   Basecbs 14490   LModclmod 17312   LSubSpclss 17378   LSpanclspn 17417  LFinGenclfig 30645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-plusg 14568  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lfig 30646
This theorem is referenced by:  islssfg  30648  lnrfg  30700
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