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Theorem islssfg2 30649
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 30648 . 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 17383 . . . . . . . . . . . 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 3511 . . . . . . . . . . 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 17431 . . . . . . . . . . 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 830 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  ( N `  b )  e.  S )  ->  (
b  C_  ( N `  b )  <->  b  C_  B ) )
13 vex 3116 . . . . . . . . . 10  |-  b  e. 
_V
1413elpw 4016 . . . . . . . . 9  |-  ( b  e.  ~P ( N `
 b )  <->  b  C_  ( N `  b ) )
1513elpw 4016 . . . . . . . . 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 2539 . . . . . . . . . 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 4013 . . . . . . . . . . 11  |-  ( ( N `  b )  =  U  ->  ~P ( N `  b )  =  ~P U )
2019eleq2d 2537 . . . . . . . . . 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 3687 . . . . . 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 2969 . 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 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   ` cfv 5588  (class class class)co 6284   Fincfn 7516   Basecbs 14490   ↾s cress 14491   LModclmod 17312   LSubSpclss 17378   LSpanclspn 17417  LFinGenclfig 30645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-sca 14571  df-vsca 14572  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lfig 30646
This theorem is referenced by:  islssfgi  30650  lsmfgcl  30652  islnm2  30656  lmhmfgima  30662
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