Theorem lspss 17191

Description: Span preserves subset ordering. (spanss 24923 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)

Hypotheses

Ref	Expression
lspss.v	|- V = ( Base ` W )
lspss.n	|- N = ( LSpan ` W )

Assertion

Ref	Expression
lspss	|- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )

Proof of Theorem lspss

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl3 993	. . . . 5 |- ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> T C_ U )
2		sstr2 3474	. . . . 5 |- ( T C_ U -> ( U C_ t -> T C_ t ) )
3	1, 2	syl 16	. . . 4 |- ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> ( U C_ t -> T C_ t ) )
4	3	ss2rabdv 3544	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> { t e. ( LSubSp ` W ) | U C_ t } C_ { t e. ( LSubSp ` W ) | T C_ t } )
5		intss 4260	. . 3 |- ( { t e. ( LSubSp ` W ) | U C_ t } C_ { t e. ( LSubSp ` W ) | T C_ t } -> |^| { t e. ( LSubSp ` W ) | T C_ t } C_ |^| { t e. ( LSubSp ` W ) | U C_ t } )
6	4, 5	syl 16	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> |^| { t e. ( LSubSp ` W ) | T C_ t } C_ |^| { t e. ( LSubSp ` W ) | U C_ t } )
7		simp1 988	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> W e. LMod )
8		simp3 990	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ U )
9		simp2 989	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> U C_ V )
10	8, 9	sstrd 3477	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ V )
11		lspss.v	. . . 4 |- V = ( Base ` W )
12		eqid 2454	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
13		lspss.n	. . . 4 |- N = ( LSpan ` W )
14	11, 12, 13	lspval 17182	. . 3 |- ( ( W e. LMod /\ T C_ V ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
15	7, 10, 14	syl2anc 661	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
16	11, 12, 13	lspval 17182	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
17	16	3adant3 1008	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
18	6, 15, 17	3sstr4d 3510	1 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   {crab 2803   C_ wss 3439   |^|cint 4239   ` cfv 5529   Basecbs 14295   LModclmod 17074   LSubSpclss 17139   LSpanclspn 17178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-0g 14502  df-mnd 15537  df-grp 15667  df-lmod 17076  df-lss 17140  df-lsp 17179

This theorem is referenced by:  lspun  17194  lspssp  17195  lspprid1  17204  lbspss  17289  lspsolvlem  17349  lspsolv  17350  lsppratlem3  17356  lbsextlem2  17366  lbsextlem3  17367  lbsextlem4  17368  lindfrn  18378  f1lindf  18379  lssats  33015  lpssat  33016  lssatle  33018  lssat  33019  dvhdimlem  35447  dvh3dim3N  35452  mapdindp2  35724  lspindp5  35773