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Theorem islssfg 27271
Description: Property of a finitely generated left (sub-)module. (Contributed by Stefan O'Rear, 1-Jan-2015.)
Hypotheses
Ref Expression
islssfg.x  |-  X  =  ( Ws  U )
islssfg.s  |-  S  =  ( LSubSp `  W )
islssfg.n  |-  N  =  ( LSpan `  W )
Assertion
Ref Expression
islssfg  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ~P  U ( b  e. 
Fin  /\  ( N `  b )  =  U ) ) )
Distinct variable groups:    W, b    X, b    S, b    U, b    N, b

Proof of Theorem islssfg
StepHypRef Expression
1 eqid 2296 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
2 islssfg.s . . . . . . 7  |-  S  =  ( LSubSp `  W )
31, 2lssss 15710 . . . . . 6  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
4 islssfg.x . . . . . . 7  |-  X  =  ( Ws  U )
54, 1ressbas2 13215 . . . . . 6  |-  ( U 
C_  ( Base `  W
)  ->  U  =  ( Base `  X )
)
63, 5syl 15 . . . . 5  |-  ( U  e.  S  ->  U  =  ( Base `  X
) )
76pweqd 3643 . . . 4  |-  ( U  e.  S  ->  ~P U  =  ~P ( Base `  X ) )
87rexeqdv 2756 . . 3  |-  ( U  e.  S  ->  ( E. b  e.  ~P  U ( b  e. 
Fin  /\  ( ( LSpan `  X ) `  b )  =  (
Base `  X )
)  <->  E. b  e.  ~P  ( Base `  X )
( b  e.  Fin  /\  ( ( LSpan `  X
) `  b )  =  ( Base `  X
) ) ) )
98adantl 452 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( E. b  e.  ~P  U ( b  e. 
Fin  /\  ( ( LSpan `  X ) `  b )  =  (
Base `  X )
)  <->  E. b  e.  ~P  ( Base `  X )
( b  e.  Fin  /\  ( ( LSpan `  X
) `  b )  =  ( Base `  X
) ) ) )
10 elpwi 3646 . . . . . 6  |-  ( b  e.  ~P U  -> 
b  C_  U )
11 islssfg.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
12 eqid 2296 . . . . . . . 8  |-  ( LSpan `  X )  =  (
LSpan `  X )
134, 11, 12, 2lsslsp 15788 . . . . . . 7  |-  ( ( W  e.  LMod  /\  U  e.  S  /\  b  C_  U )  ->  ( N `  b )  =  ( ( LSpan `  X ) `  b
) )
14133expa 1151 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  b  C_  U
)  ->  ( N `  b )  =  ( ( LSpan `  X ) `  b ) )
1510, 14sylan2 460 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  b  e.  ~P U )  ->  ( N `  b )  =  ( ( LSpan `  X ) `  b
) )
166ad2antlr 707 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  b  e.  ~P U )  ->  U  =  ( Base `  X
) )
1715, 16eqeq12d 2310 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  b  e.  ~P U )  ->  (
( N `  b
)  =  U  <->  ( ( LSpan `  X ) `  b )  =  (
Base `  X )
) )
1817anbi2d 684 . . 3  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  b  e.  ~P U )  ->  (
( b  e.  Fin  /\  ( N `  b
)  =  U )  <-> 
( b  e.  Fin  /\  ( ( LSpan `  X
) `  b )  =  ( Base `  X
) ) ) )
1918rexbidva 2573 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( E. b  e.  ~P  U ( b  e. 
Fin  /\  ( N `  b )  =  U )  <->  E. b  e.  ~P  U ( b  e. 
Fin  /\  ( ( LSpan `  X ) `  b )  =  (
Base `  X )
) ) )
204, 2lsslmod 15733 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
21 eqid 2296 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
2221, 12islmodfg 27270 . . 3  |-  ( X  e.  LMod  ->  ( X  e. LFinGen 
<->  E. b  e.  ~P  ( Base `  X )
( b  e.  Fin  /\  ( ( LSpan `  X
) `  b )  =  ( Base `  X
) ) ) )
2320, 22syl 15 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ~P  ( Base `  X )
( b  e.  Fin  /\  ( ( LSpan `  X
) `  b )  =  ( Base `  X
) ) ) )
249, 19, 233bitr4rd 277 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ~P  U ( b  e. 
Fin  /\  ( N `  b )  =  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ~Pcpw 3638   ` cfv 5271  (class class class)co 5874   Fincfn 6879   Basecbs 13164   ↾s cress 13165   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744  LFinGenclfig 27268
This theorem is referenced by:  islssfg2  27272  lmhmfgsplit  27287
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-3 9821  df-4 9822  df-5 9823  df-6 9824  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  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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