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Theorem islmodfg 29448
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 29447 . . . 4  |- LFinGen  =  {
a  e.  LMod  |  (
Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) }
21eleq2i 2507 . . 3  |-  ( W  e. LFinGen 
<->  W  e.  { a  e.  LMod  |  ( Base `  a )  e.  ( ( LSpan `  a
) " ( ~P ( Base `  a
)  i^i  Fin )
) } )
3 fveq2 5712 . . . . 5  |-  ( a  =  W  ->  ( Base `  a )  =  ( Base `  W
) )
4 fveq2 5712 . . . . . . 7  |-  ( a  =  W  ->  ( LSpan `  a )  =  ( LSpan `  W )
)
5 islmodfg.n . . . . . . 7  |-  N  =  ( LSpan `  W )
64, 5syl6eqr 2493 . . . . . 6  |-  ( a  =  W  ->  ( LSpan `  a )  =  N )
73pweqd 3886 . . . . . . 7  |-  ( a  =  W  ->  ~P ( Base `  a )  =  ~P ( Base `  W
) )
87ineq1d 3572 . . . . . 6  |-  ( a  =  W  ->  ( ~P ( Base `  a
)  i^i  Fin )  =  ( ~P ( Base `  W )  i^i 
Fin ) )
96, 8imaeq12d 5191 . . . . 5  |-  ( a  =  W  ->  (
( LSpan `  a ) " ( ~P ( Base `  a )  i^i 
Fin ) )  =  ( N " ( ~P ( Base `  W
)  i^i  Fin )
) )
103, 9eleq12d 2511 . . . 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 3139 . . 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 2443 . . . . . 6  |-  ( Base `  W )  =  (
Base `  W )
14 eqid 2443 . . . . . 6  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 5lspf 17077 . . . . 5  |-  ( W  e.  LMod  ->  N : ~P ( Base `  W
) --> ( LSubSp `  W
) )
16 ffn 5580 . . . . 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 3591 . . . 4  |-  ( ~P ( Base `  W
)  i^i  Fin )  C_ 
~P ( Base `  W
)
19 fvelimab 5768 . . . 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 3560 . . . . . . 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 2447 . . . . . . . . . 10  |-  ( Base `  W )  =  B
2423pweqi 3885 . . . . . . . . 9  |-  ~P ( Base `  W )  =  ~P B
2524eleq2i 2507 . . . . . . . 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 2453 . . . . . 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 2765 . . 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 1369    e. wcel 1756   E.wrex 2737   {crab 2740    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439   Fincfn 7331   Basecbs 14195   LModclmod 16970   LSubSpclss 17035   LSpanclspn 17074  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-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  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-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-plusg 14272  df-0g 14401  df-mnd 15436  df-grp 15566  df-minusg 15567  df-sbg 15568  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lfig 29447
This theorem is referenced by:  islssfg  29449  lnrfg  29501
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