Theorem aspss 15002

Description: Span preserves subset ordering. (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.

Step	Hyp	Ref	Expression
1		simpl3 1033	. . . . 5 |- ( ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) /\ t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) ) -> T C_ S )
2		sstr2 3255	. . . . 5 |- ( T C_ S -> ( S C_ t -> T C_ t ) )
3	1, 2	syl 14	. . . 4 |- ( ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) /\ t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) ) -> ( S C_ t -> T C_ t ) )
4	3	ss2rabdv 3329	. . 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 3989	. . 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 } )
6	4, 5	syl 14	. 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 1028	. . 3 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> W e. AssAlg )
8		simp3 1030	. . . 4 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> T C_ S )
9		simp2 1029	. . . 4 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> S C_ V )
10	8, 9	sstrd 3258	. . 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 2238	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
14	11, 12, 13	aspval 14998	. . 3 |- ( ( W e. AssAlg /\ T C_ V ) -> ( A ` T ) = |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } )
15	7, 10, 14	syl2anc 415	. 2 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) = |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | T C_ t } )
16	11, 12, 13	aspval 14998	. . 3 |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
17	16	3adant3 1048	. 2 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` S ) = |^| { t e. ( (SubRing ` W ) i^i ( LSubSp ` W ) ) | S C_ t } )
18	6, 15, 17	3sstr4d 3293	1 |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) C_ ( A ` S ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   i^i cin 3219   C_ wss 3220   |^|cint 3968   ` cfv 5375   Basecbs 13335  SubRingcsubrg 14508   LSubSpclss 14672  AssAlgcasa 14979  AlgSpancasp 14980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-addass 8275  ax-i2m1 8278  ax-0lt1 8279  ax-0id 8281  ax-rnegex 8282  ax-pre-ltirr 8285  ax-pre-lttrn 8287  ax-pre-ltadd 8289

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-pnf 8356  df-mnf 8357  df-ltxr 8359  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-ndx 13338  df-slot 13339  df-base 13341  df-sets 13342  df-iress 13343  df-plusg 13427  df-mulr 13428  df-sca 13430  df-vsca 13431  df-0g 13595  df-mgm 13659  df-sgrp 13700  df-mnd 13713  df-grp 13791  df-minusg 13792  df-mgp 14201  df-ur 14246  df-ring 14285  df-subrg 14510  df-lmod 14608  df-lssm 14673  df-assa 14982  df-asp 14983

This theorem is referenced by: (None)