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Theorem lsslss 16037
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 16036 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  X  e.  LMod )
4 eqid 2436 . . . 4  |-  ( Xs  V )  =  ( Xs  V )
5 eqid 2436 . . . 4  |-  ( Base `  X )  =  (
Base `  X )
6 lsslss.t . . . 4  |-  T  =  ( LSubSp `  X )
74, 5, 6islss3 16035 . . 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 2436 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
109, 2lssss 16013 . . . . . 6  |-  ( U  e.  S  ->  U  C_  ( Base `  W
) )
1110adantl 453 . . . . 5  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  C_  ( Base `  W
) )
121, 9ressbas2 13520 . . . . 5  |-  ( U 
C_  ( Base `  W
)  ->  U  =  ( Base `  X )
)
1311, 12syl 16 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  U  =  ( Base `  X
) )
1413sseq2d 3376 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  C_  U  <->  V  C_  ( Base `  X ) ) )
1514anbi1d 686 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  C_  ( Base `  X )  /\  ( Xs  V )  e.  LMod ) ) )
16 sstr2 3355 . . . . . . 7  |-  ( V 
C_  U  ->  ( U  C_  ( Base `  W
)  ->  V  C_  ( Base `  W ) ) )
1711, 16mpan9 456 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  V  C_  ( Base `  W ) )
1817biantrurd 495 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Ws  V )  e.  LMod  <->  ( V  C_  ( Base `  W
)  /\  ( Ws  V
)  e.  LMod )
) )
191oveq1i 6091 . . . . . . 7  |-  ( Xs  V )  =  ( ( Ws  U )s  V )
20 ressabs 13527 . . . . . . . 8  |-  ( ( U  e.  S  /\  V  C_  U )  -> 
( ( Ws  U )s  V )  =  ( Ws  V ) )
2120adantll 695 . . . . . . 7  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Ws  U )s  V )  =  ( Ws  V ) )
2219, 21syl5eq 2480 . . . . . 6  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( Xs  V
)  =  ( Ws  V ) )
2322eleq1d 2502 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Xs  V )  e.  LMod  <->  ( Ws  V )  e.  LMod ) )
24 eqid 2436 . . . . . . 7  |-  ( Ws  V )  =  ( Ws  V )
2524, 9, 2islss3 16035 . . . . . 6  |-  ( W  e.  LMod  ->  ( V  e.  S  <->  ( V  C_  ( Base `  W
)  /\  ( Ws  V
)  e.  LMod )
) )
2625ad2antrr 707 . . . . 5  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( V  e.  S  <->  ( V  C_  ( Base `  W )  /\  ( Ws  V )  e.  LMod ) ) )
2718, 23, 263bitr4d 277 . . . 4  |-  ( ( ( W  e.  LMod  /\  U  e.  S )  /\  V  C_  U
)  ->  ( ( Xs  V )  e.  LMod  <->  V  e.  S ) )
2827pm5.32da 623 . . 3  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  C_  U  /\  V  e.  S
) ) )
29 ancom 438 . . 3  |-  ( ( V  C_  U  /\  V  e.  S )  <->  ( V  e.  S  /\  V  C_  U ) )
3028, 29syl6bb 253 . 2  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  (
( V  C_  U  /\  ( Xs  V )  e.  LMod ) 
<->  ( V  e.  S  /\  V  C_  U ) ) )
318, 15, 303bitr2d 273 1  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  ( V  e.  T  <->  ( V  e.  S  /\  V  C_  U ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   ↾s cress 13470   LModclmod 15950   LSubSpclss 16008
This theorem is referenced by:  lsslsp  16091  mplbas2  16531  mplind  16562  lnmlsslnm  27156  lmhmlnmsplit  27162  lcdlss  32417
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009
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