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Theorem islmodfg 35390
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 35389 . . . 4  |- LFinGen  =  {
a  e.  LMod  |  (
Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) }
21eleq2i 2482 . . 3  |-  ( W  e. LFinGen 
<->  W  e.  { a  e.  LMod  |  ( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) } )
3 fveq2 5851 . . . . 5  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
4 fveq2 5851 . . . . . . 7  |-  ( a  =  W  ->  ( LSpan `  a )  =  ( LSpan `  W )
)
5 islmodfg.n . . . . . . 7  |-  N  =  ( LSpan `  W )
64, 5syl6eqr 2463 . . . . . 6  |-  ( a  =  W  ->  ( LSpan `  a )  =  N )
73pweqd 3962 . . . . . . 7  |-  ( a  =  W  ->  ~P ( Base `  a )  =  ~P ( Base `  W
) )
87ineq1d 3642 . . . . . 6  |-  ( a  =  W  ->  ( ~P ( Base `  a
)  i^i  Fin )  =  ( ~P ( Base `  W )  i^i 
Fin ) )
96, 8imaeq12d 5160 . . . . 5  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
103, 9eleq12d 2486 . . . 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 3210 . . 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 259 . 2  |-  ( W  e.  LMod  ->  ( W  e. LFinGen 
<->  ( Base `  W
)  e.  ( N
" ( ~P ( Base `  W )  i^i 
Fin ) ) ) )
13 eqid 2404 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
14 eqid 2404 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 5lspf 17942 . . . . 5  |-  ( W  e.  LMod  ->  N : ~P ( Base `  W
) --> ( LSubSp `  W
) )
16 ffn 5716 . . . . 5  |-  ( N : ~P ( Base `  W ) --> ( LSubSp `  W )  ->  N  Fn  ~P ( Base `  W
) )
1715, 16syl 17 . . . 4  |-  ( W  e.  LMod  ->  N  Fn  ~P ( Base `  W
) )
18 inss1 3661 . . . 4  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
19 fvelimab 5907 . . . 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 3628 . . . . . . 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 2417 . . . . . . . . . 10  |-  ( Base `  W )  =  B
2423pweqi 3961 . . . . . . . . 9  |-  ~P ( Base `  W )  =  ~P B
2524eleq2i 2482 . . . . . . . 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 251 . . . . . 6  |-  ( b  e.  ( ~P ( Base `  W )  i^i 
Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2823eqeq2i 2422 . . . . . 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 251 . . . 4  |-  ( ( b  e.  ( ~P ( Base `  W
)  i^i  Fin )  /\  ( N `  b
)  =  ( Base `  W ) )  <->  ( b  e.  ~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  B ) ) )
3231rexbii2 2906 . . 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 263 . 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 255 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 186    /\ wa 369    = wceq 1407    e. wcel 1844   E.wrex 2757   {crab 2760    i^i cin 3415    C_ wss 3416   ~Pcpw 3957   "cima 4828    Fn wfn 5566   -->wf 5567   ` cfv 5571   Fincfn 7556   Basecbs 14843   LModclmod 17834   LSubSpclss 17900   LSpanclspn 17939  LFinGenclfig 35388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-plusg 14924  df-0g 15058  df-mgm 16198  df-sgrp 16237  df-mnd 16247  df-grp 16383  df-minusg 16384  df-sbg 16385  df-mgp 17464  df-ur 17476  df-ring 17522  df-lmod 17836  df-lss 17901  df-lsp 17940  df-lfig 35389
This theorem is referenced by:  islssfg  35391  lnrfg  35445
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