Theorem lspss 14547

Description: Span preserves subset ordering. (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 1029	. . . . 5 |- ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> T C_ U )
2		sstr2 3245	. . . . 5 |- ( T C_ U -> ( U C_ t -> T C_ t ) )
3	1, 2	syl 14	. . . 4 |- ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> ( U C_ t -> T C_ t ) )
4	3	ss2rabdv 3319	. . 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 3970	. . 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 14	. 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 1024	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> W e. LMod )
8		simp3 1026	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ U )
9		simp2 1025	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> U C_ V )
10	8, 9	sstrd 3248	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ V )
11		lspss.v	. . . 4 |- V = ( Base ` W )
12		eqid 2232	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
13		lspss.n	. . . 4 |- N = ( LSpan ` W )
14	11, 12, 13	lspval 14538	. . 3 |- ( ( W e. LMod /\ T C_ V ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
15	7, 10, 14	syl2anc 411	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) = |^| { t e. ( LSubSp ` W ) | T C_ t } )
16	11, 12, 13	lspval 14538	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
17	16	3adant3 1044	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
18	6, 15, 17	3sstr4d 3283	1 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   {crab 2524   C_ wss 3211   |^|cint 3949   ` cfv 5352   Basecbs 13212   LModclmod 14435   LSubSpclss 14500   LSpanclspn 14534

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-ndx 13215  df-slot 13216  df-base 13218  df-plusg 13303  df-mulr 13304  df-sca 13306  df-vsca 13307  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-grp 13716  df-lmod 14437  df-lssm 14501  df-lsp 14535

This theorem is referenced by:  lspun  14550  lspssp  14551  lspprid1  14559