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Theorem aspss 17849
Description: Span preserves subset ordering. (spanss 26131 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 1000 . . . . 5  |-  ( ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S )  /\  t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) ) )  ->  T  C_  S
)
2 sstr2 3493 . . . . 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 3563 . . 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 4289 . . 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 995 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  W  e. AssAlg )
8 simp3 997 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  T  C_  S
)
9 simp2 996 . . . 4  |-  ( ( W  e. AssAlg  /\  S  C_  V  /\  T  C_  S
)  ->  S  C_  V
)
108, 9sstrd 3496 . . 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 2441 . . . 4  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1411, 12, 13aspval 17845 . . 3  |-  ( ( W  e. AssAlg  /\  T  C_  V )  ->  ( A `  T )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  T  C_  t }
)
157, 10, 14syl2anc 661 . 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 17845 . . 3  |-  ( ( W  e. AssAlg  /\  S  C_  V )  ->  ( A `  S )  =  |^| { t  e.  ( (SubRing `  W
)  i^i  ( LSubSp `  W ) )  |  S  C_  t }
)
17163adant3 1015 . 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 3529 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 369    /\ w3a 972    = wceq 1381    e. wcel 1802   {crab 2795    i^i cin 3457    C_ wss 3458   |^|cint 4267   ` cfv 5574   Basecbs 14504  SubRingcsubrg 17293   LSubSpclss 17446  AssAlgcasa 17826  AlgSpancasp 17827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  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 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-ndx 14507  df-slot 14508  df-base 14509  df-sets 14510  df-ress 14511  df-plusg 14582  df-0g 14711  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-mgp 17010  df-ur 17022  df-ring 17068  df-subrg 17295  df-lmod 17382  df-lss 17447  df-assa 17829  df-asp 17830
This theorem is referenced by:  mplbas2  18002  mplbas2OLD  18003  mplind  18035
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