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Theorem islssfg2 29421
Description: Property of a finitely generated left (sub-)module, with a relaxed constraint on the spanning vectors. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
islssfg.x  |-  X  =  ( Ws  U )
islssfg.s  |-  S  =  ( LSubSp `  W )
islssfg.n  |-  N  =  ( LSpan `  W )
islssfg2.b  |-  B  =  ( Base `  W
)
Assertion
Ref Expression
islssfg2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ( ~P B  i^i  Fin ) ( N `  b )  =  U ) )
Distinct variable groups:    W, b    X, b    S, b    U, b    N, b
Allowed substitution hint:    B( b)

Proof of Theorem islssfg2
StepHypRef Expression
1 islssfg.x . . 3  |-  X  =  ( Ws  U )
2 islssfg.s . . 3  |-  S  =  ( LSubSp `  W )
3 islssfg.n . . 3  |-  N  =  ( LSpan `  W )
41, 2, 3islssfg 29420 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ~P  U ( b  e. 
Fin  /\  ( N `  b )  =  U ) ) )
5 islssfg2.b . . . . . . . . . . . . 13  |-  B  =  ( Base `  W
)
65, 2lssss 17016 . . . . . . . . . . . 12  |-  ( ( N `  b )  e.  S  ->  ( N `  b )  C_  B )
76adantl 466 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  ( N `  b )  C_  B )
8 sstr2 3361 . . . . . . . . . . 11  |-  ( b 
C_  ( N `  b )  ->  (
( N `  b
)  C_  B  ->  b 
C_  B ) )
97, 8mpan9 469 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  ( N `  b
)  e.  S )  /\  b  C_  ( N `  b )
)  ->  b  C_  B )
105, 3lspssid 17064 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  b  C_  B )  ->  b  C_  ( N `  b
) )
1110adantlr 714 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  ( N `  b
)  e.  S )  /\  b  C_  B
)  ->  b  C_  ( N `  b ) )
129, 11impbida 828 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  (
b  C_  ( N `  b )  <->  b  C_  B ) )
13 vex 2973 . . . . . . . . . 10  |-  b  e. 
_V
1413elpw 3864 . . . . . . . . 9  |-  ( b  e.  ~P ( N `
 b )  <->  b  C_  ( N `  b ) )
1513elpw 3864 . . . . . . . . 9  |-  ( b  e.  ~P B  <->  b  C_  B )
1612, 14, 153bitr4g 288 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  (
b  e.  ~P ( N `  b )  <->  b  e.  ~P B ) )
17 eleq1 2501 . . . . . . . . . 10  |-  ( ( N `  b )  =  U  ->  (
( N `  b
)  e.  S  <->  U  e.  S ) )
1817anbi2d 703 . . . . . . . . 9  |-  ( ( N `  b )  =  U  ->  (
( W  e.  LMod  /\  ( N `  b
)  e.  S )  <-> 
( W  e.  LMod  /\  U  e.  S ) ) )
19 pweq 3861 . . . . . . . . . . 11  |-  ( ( N `  b )  =  U  ->  ~P ( N `  b )  =  ~P U )
2019eleq2d 2508 . . . . . . . . . 10  |-  ( ( N `  b )  =  U  ->  (
b  e.  ~P ( N `  b )  <->  b  e.  ~P U ) )
2120bibi1d 319 . . . . . . . . 9  |-  ( ( N `  b )  =  U  ->  (
( b  e.  ~P ( N `  b )  <-> 
b  e.  ~P B
)  <->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2218, 21imbi12d 320 . . . . . . . 8  |-  ( ( N `  b )  =  U  ->  (
( ( W  e. 
LMod  /\  ( N `  b )  e.  S
)  ->  ( b  e.  ~P ( N `  b )  <->  b  e.  ~P B ) )  <->  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
b  e.  ~P U  <->  b  e.  ~P B ) ) ) )
2316, 22mpbii 211 . . . . . . 7  |-  ( ( N `  b )  =  U  ->  (
( W  e.  LMod  /\  U  e.  S )  ->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2423com12 31 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( N `  b
)  =  U  -> 
( b  e.  ~P U 
<->  b  e.  ~P B
) ) )
2524adantld 467 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( b  e.  Fin  /\  ( N `  b
)  =  U )  ->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2625pm5.32rd 640 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( b  e.  ~P U  /\  ( b  e. 
Fin  /\  ( N `  b )  =  U ) )  <->  ( b  e.  ~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  U ) ) ) )
27 elin 3537 . . . . . 6  |-  ( b  e.  ( ~P B  i^i  Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2827anbi1i 695 . . . . 5  |-  ( ( b  e.  ( ~P B  i^i  Fin )  /\  ( N `  b
)  =  U )  <-> 
( ( b  e. 
~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  U ) )
29 anass 649 . . . . 5  |-  ( ( ( b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  U )  <->  ( b  e. 
~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  U ) ) )
3028, 29bitr2i 250 . . . 4  |-  ( ( b  e.  ~P B  /\  ( b  e.  Fin  /\  ( N `  b
)  =  U ) )  <->  ( b  e.  ( ~P B  i^i  Fin )  /\  ( N `
 b )  =  U ) )
3126, 30syl6bb 261 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( b  e.  ~P U  /\  ( b  e. 
Fin  /\  ( N `  b )  =  U ) )  <->  ( b  e.  ( ~P B  i^i  Fin )  /\  ( N `
 b )  =  U ) ) )
3231rexbidv2 2736 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( E. b  e.  ~P  U ( b  e. 
Fin  /\  ( N `  b )  =  U )  <->  E. b  e.  ( ~P B  i^i  Fin ) ( N `  b )  =  U ) )
334, 32bitrd 253 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( X  e. LFinGen  <->  E. b  e.  ( ~P B  i^i  Fin ) ( N `  b )  =  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   E.wrex 2714    i^i cin 3325    C_ wss 3326   ~Pcpw 3858   ` cfv 5416  (class class class)co 6089   Fincfn 7308   Basecbs 14172   ↾s cress 14173   LModclmod 16946   LSubSpclss 17011   LSpanclspn 17050  LFinGenclfig 29417
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 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-4 10380  df-5 10381  df-6 10382  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-sca 14252  df-vsca 14253  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-sbg 15545  df-subg 15676  df-mgp 16590  df-ur 16602  df-rng 16645  df-lmod 16948  df-lss 17012  df-lsp 17051  df-lfig 29418
This theorem is referenced by:  islssfgi  29422  lsmfgcl  29424  islnm2  29428  lmhmfgima  29434
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