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Theorem islssfg2 27127
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 27126 . 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 16005 . . . . . . . . . . . 12  |-  ( ( N `  b )  e.  S  ->  ( N `  b )  C_  B )
76adantl 453 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  ( N `  b )  C_  B )
8 sstr2 3347 . . . . . . . . . . 11  |-  ( b 
C_  ( N `  b )  ->  (
( N `  b
)  C_  B  ->  b 
C_  B ) )
97, 8mpan9 456 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  ( N `  b
)  e.  S )  /\  b  C_  ( N `  b )
)  ->  b  C_  B )
105, 3lspssid 16053 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  b  C_  B )  ->  b  C_  ( N `  b
) )
1110adantlr 696 . . . . . . . . . 10  |-  ( ( ( W  e.  LMod  /\  ( N `  b
)  e.  S )  /\  b  C_  B
)  ->  b  C_  ( N `  b ) )
129, 11impbida 806 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  (
b  C_  ( N `  b )  <->  b  C_  B ) )
13 vex 2951 . . . . . . . . . 10  |-  b  e. 
_V
1413elpw 3797 . . . . . . . . 9  |-  ( b  e.  ~P ( N `
 b )  <->  b  C_  ( N `  b ) )
1513elpw 3797 . . . . . . . . 9  |-  ( b  e.  ~P B  <->  b  C_  B )
1612, 14, 153bitr4g 280 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  (
b  e.  ~P ( N `  b )  <->  b  e.  ~P B ) )
17 eleq1 2495 . . . . . . . . . 10  |-  ( ( N `  b )  =  U  ->  (
( N `  b
)  e.  S  <->  U  e.  S ) )
1817anbi2d 685 . . . . . . . . 9  |-  ( ( N `  b )  =  U  ->  (
( W  e.  LMod  /\  ( N `  b
)  e.  S )  <-> 
( W  e.  LMod  /\  U  e.  S ) ) )
19 pweq 3794 . . . . . . . . . . 11  |-  ( ( N `  b )  =  U  ->  ~P ( N `  b )  =  ~P U )
2019eleq2d 2502 . . . . . . . . . 10  |-  ( ( N `  b )  =  U  ->  (
b  e.  ~P ( N `  b )  <->  b  e.  ~P U ) )
2120bibi1d 311 . . . . . . . . 9  |-  ( ( N `  b )  =  U  ->  (
( b  e.  ~P ( N `  b )  <-> 
b  e.  ~P B
)  <->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2218, 21imbi12d 312 . . . . . . . 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 203 . . . . . . 7  |-  ( ( N `  b )  =  U  ->  (
( W  e.  LMod  /\  U  e.  S )  ->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2423com12 29 . . . . . 6  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( N `  b
)  =  U  -> 
( b  e.  ~P U 
<->  b  e.  ~P B
) ) )
2524adantld 454 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( b  e.  Fin  /\  ( N `  b
)  =  U )  ->  ( b  e. 
~P U  <->  b  e.  ~P B ) ) )
2625pm5.32rd 622 . . . 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 3522 . . . . . 6  |-  ( b  e.  ( ~P B  i^i  Fin )  <->  ( b  e.  ~P B  /\  b  e.  Fin ) )
2827anbi1i 677 . . . . 5  |-  ( ( b  e.  ( ~P B  i^i  Fin )  /\  ( N `  b
)  =  U )  <-> 
( ( b  e. 
~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  U ) )
29 anass 631 . . . . 5  |-  ( ( ( b  e.  ~P B  /\  b  e.  Fin )  /\  ( N `  b )  =  U )  <->  ( b  e. 
~P B  /\  (
b  e.  Fin  /\  ( N `  b )  =  U ) ) )
3028, 29bitr2i 242 . . . 4  |-  ( ( b  e.  ~P B  /\  ( b  e.  Fin  /\  ( N `  b
)  =  U ) )  <->  ( b  e.  ( ~P B  i^i  Fin )  /\  ( N `
 b )  =  U ) )
3126, 30syl6bb 253 . . 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 2720 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   ` cfv 5446  (class class class)co 6073   Fincfn 7101   Basecbs 13461   ↾s cress 13462   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039  LFinGenclfig 27123
This theorem is referenced by:  islssfgi  27128  lsmfgcl  27130  islnm2  27134  lmhmfgima  27140
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lfig 27124
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