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Theorem aspss 16381
Description: Span preserves subset ordering. (spanss 22840 analog.) (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
aspval.a  |-  A  =  (AlgSpan `  W )
aspval.v  |-  V  =  ( Base `  W
)
Assertion
Ref Expression
aspss  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  ( A `  T )  C_  ( A `  S )
)

Proof of Theorem aspss
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 simpl3 962 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S )  /\  t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) ) )  ->  T  C_  S
)
2 sstr2 3347 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  t  ->  T  C_  t ) )
31, 2syl 16 . . . 4  |-  ( ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S )  /\  t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) ) )  ->  ( S  C_  t  ->  T  C_  t
) )
43ss2rabdv 3416 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }  C_ 
{ t  e.  ( (SubRing `  W )  i^i  ( LSubSp `  W )
)  |  T  C_  t } )
5 intss 4063 . . 3  |-  ( { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  |  S  C_  t }  C_  { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }  C_ 
|^| { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `  W )
)  |  S  C_  t } )
64, 5syl 16 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }  C_ 
|^| { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `  W )
)  |  S  C_  t } )
7 simp1 957 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  W  e. AssAlg )
8 simp3 959 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  T  C_  S
)
9 simp2 958 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  S  C_  V
)
108, 9sstrd 3350 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  T  C_  V
)
11 aspval.a . . . 4  |-  A  =  (AlgSpan `  W )
12 aspval.v . . . 4  |-  V  =  ( Base `  W
)
13 eqid 2435 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1411, 12, 13aspval 16377 . . 3  |-  ( ( W  e. AssAlg  /\  T  C_  V )  ->  ( A `  T )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }
)
157, 10, 14syl2anc 643 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  ( A `  T )  =  |^| { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  |  T  C_  t } )
1611, 12, 13aspval 16377 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }
)
17163adant3 977 . 2  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W )  i^i  ( LSubSp `
 W ) )  |  S  C_  t } )
186, 15, 173sstr4d 3383 1  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  ( A `  T )  C_  ( A `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312   |^|cint 4042   ` cfv 5446   Basecbs 13459  SubRingcsubrg 15854   LSubSpclss 15998  AssAlgcasa 16359  AlgSpancasp 16360
This theorem is referenced by:  mplbas2  16521  mplind  16552
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 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
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-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-0g 13717  df-mnd 14680  df-grp 14802  df-mgp 15639  df-rng 15653  df-ur 15655  df-subrg 15856  df-lmod 15942  df-lss 15999  df-assa 16362  df-asp 16363
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