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Theorem aspss 18499
Description: Span preserves subset ordering. (spanss 26943 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 1010 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S )  /\  t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) ) )  ->  T  C_  S
)
2 sstr2 3414 . . . . 5  |-  ( T 
C_  S  ->  ( S  C_  t  ->  T  C_  t ) )
31, 2syl 17 . . . 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 3485 . . 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 4219 . . 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 17 . 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 1005 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  W  e. AssAlg )
8 simp3 1007 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  T  C_  S
)
9 simp2 1006 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  S  C_  V
)
108, 9sstrd 3417 . . 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 2428 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1411, 12, 13aspval 18495 . . 3  |-  ( ( W  e. AssAlg  /\  T  C_  V )  ->  ( A `  T )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }
)
157, 10, 14syl2anc 665 . 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 18495 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }
)
17163adant3 1025 . 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 3450 1  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  ( A `  T )  C_  ( A `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {crab 2718    i^i cin 3378    C_ wss 3379   |^|cint 4198   ` cfv 5544   Basecbs 15064  SubRingcsubrg 17947   LSubSpclss 18098  AssAlgcasa 18476  AlgSpancasp 18477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-0g 15283  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-grp 16616  df-mgp 17667  df-ur 17679  df-ring 17725  df-subrg 17949  df-lmod 18036  df-lss 18099  df-assa 18479  df-asp 18480
This theorem is referenced by:  mplbas2  18637  mplind  18668
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