Theorem lspss 14478

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 3235	. . . . 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 3309	. . 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 3954	. . 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 3238	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ V )
11		lspss.v	. . . 4 |- V = ( Base ` W )
12		eqid 2231	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
13		lspss.n	. . . 4 |- N = ( LSpan ` W )
14	11, 12, 13	lspval 14469	. . 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 14469	. . 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 3273	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 2202   {crab 2515   C_ wss 3201   |^|cint 3933   ` cfv 5333   Basecbs 13145   LModclmod 14366   LSubSpclss 14431   LSpanclspn 14465

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-ndx 13148  df-slot 13149  df-base 13151  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-0g 13404  df-mgm 13502  df-sgrp 13548  df-mnd 13563  df-grp 13649  df-lmod 14368  df-lssm 14432  df-lsp 14466

This theorem is referenced by:  lspun  14481  lspssp  14482  lspprid1  14490