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Theorem lsslss 17040
Description: The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
Hypotheses
Ref Expression
lsslss.x  |-  X  =  ( Ws  U )
lsslss.s  |-  S  =  ( LSubSp `  W )
lsslss.t  |-  T  =  ( LSubSp `  X )
Assertion
Ref Expression
lsslss  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )

Proof of Theorem lsslss
StepHypRef Expression
1 lsslss.x . . . 4  |-  X  =  ( Ws  U )
2 lsslss.s . . . 4  |-  S  =  ( LSubSp `  W )
31, 2lsslmod 17039 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
4 eqid 2441 . . . 4  |-  ( Xs  V )  =  ( Xs  V )
5 eqid 2441 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
6 lsslss.t . . . 4  |-  T  =  ( LSubSp `  X )
74, 5, 6islss3 17038 . . 3  |-  ( X  e.  LMod  ->  ( V  e.  T  <->  ( V  C_  ( Base `  X
)  /\  ( Xs  V
)  e.  LMod )
) )
83, 7syl 16 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  e.  T  <->  ( V  C_  ( Base `  X
)  /\  ( Xs  V
)  e.  LMod )
) )
9 eqid 2441 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
109, 2lssss 17016 . . . . . 6  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
1110adantl 466 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  ( Base `  W
) )
121, 9ressbas2 14227 . . . . 5  |-  ( U 
C_  ( Base `  W
)  ->  U  =  ( Base `  X )
)
1311, 12syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
1413sseq2d 3382 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  C_  U  <->  V  C_  ( Base `  X ) ) )
1514anbi1d 704 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  C_  ( Base `  X )  /\  ( Xs  V )  e.  LMod ) ) )
16 sstr2 3361 . . . . . . 7  |-  ( V 
C_  U  ->  ( U  C_  ( Base `  W
)  ->  V  C_  ( Base `  W ) ) )
1711, 16mpan9 469 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  V  C_  ( Base `  W ) )
1817biantrurd 508 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Ws  V )  e.  LMod  <->  ( V  C_  ( Base `  W
)  /\  ( Ws  V
)  e.  LMod )
) )
191oveq1i 6099 . . . . . . 7  |-  ( Xs  V )  =  ( ( Ws  U )s  V )
20 ressabs 14234 . . . . . . . 8  |-  ( ( U  e.  S  /\  V  C_  U )  -> 
( ( Ws  U )s  V )  =  ( Ws  V ) )
2120adantll 713 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Ws  U )s  V )  =  ( Ws  V ) )
2219, 21syl5eq 2485 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( Xs  V
)  =  ( Ws  V ) )
2322eleq1d 2507 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Xs  V )  e.  LMod  <->  ( Ws  V )  e.  LMod ) )
24 eqid 2441 . . . . . . 7  |-  ( Ws  V )  =  ( Ws  V )
2524, 9, 2islss3 17038 . . . . . 6  |-  ( W  e.  LMod  ->  ( V  e.  S  <->  ( V  C_  ( Base `  W
)  /\  ( Ws  V
)  e.  LMod )
) )
2625ad2antrr 725 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( V  e.  S  <->  ( V  C_  ( Base `  W )  /\  ( Ws  V )  e.  LMod ) ) )
2718, 23, 263bitr4d 285 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Xs  V )  e.  LMod  <->  V  e.  S ) )
2827pm5.32da 641 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  C_  U  /\  V  e.  S
) ) )
29 ancom 450 . . 3  |-  ( ( V  C_  U  /\  V  e.  S )  <->  ( V  e.  S  /\  V  C_  U ) )
3028, 29syl6bb 261 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  e.  S  /\  V  C_  U ) ) )
318, 15, 303bitr2d 281 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3326   ` cfv 5416  (class class class)co 6089   Basecbs 14172   ↾s cress 14173   LModclmod 16946   LSubSpclss 17011
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-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
This theorem is referenced by:  lsslsp  17094  mplbas2  17549  mplbas2OLD  17550  mplind  17582  lnmlsslnm  29431  lmhmlnmsplit  29437  lcdlss  35261
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