Theorem lspss 14378

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 1026	. . . . 5 |- ( ( ( W e. LMod /\ U C_ V /\ T C_ U ) /\ t e. ( LSubSp ` W ) ) -> T C_ U )
2		sstr2 3231	. . . . 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 3305	. . 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 3944	. . 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 1021	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> W e. LMod )
8		simp3 1023	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ U )
9		simp2 1022	. . . 4 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> U C_ V )
10	8, 9	sstrd 3234	. . 3 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> T C_ V )
11		lspss.v	. . . 4 |- V = ( Base ` W )
12		eqid 2229	. . . 4 |- ( LSubSp ` W ) = ( LSubSp ` W )
13		lspss.n	. . . 4 |- N = ( LSpan ` W )
14	11, 12, 13	lspval 14369	. . 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 14369	. . 3 |- ( ( W e. LMod /\ U C_ V ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
17	16	3adant3 1041	. 2 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` U ) = |^| { t e. ( LSubSp ` W ) | U C_ t } )
18	6, 15, 17	3sstr4d 3269	1 |- ( ( W e. LMod /\ U C_ V /\ T C_ U ) -> ( N ` T ) C_ ( N ` U ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   C_ wss 3197   |^|cint 3923   ` cfv 5318   Basecbs 13047   LModclmod 14266   LSubSpclss 14331   LSpanclspn 14365

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8101  ax-resscn 8102  ax-1re 8104  ax-addrcl 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-ndx 13050  df-slot 13051  df-base 13053  df-plusg 13138  df-mulr 13139  df-sca 13141  df-vsca 13142  df-0g 13306  df-mgm 13404  df-sgrp 13450  df-mnd 13465  df-grp 13551  df-lmod 14268  df-lssm 14332  df-lsp 14366

This theorem is referenced by:  lspun  14381  lspssp  14382  lspprid1  14390