Theorem lspun0 14590

Description: The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)

Hypotheses

Ref	Expression
lspun0.v	|- V = ( Base ` W )
lspun0.o	|- .0. = ( 0g ` W )
lspun0.n	|- N = ( LSpan ` W )
lspun0.w	|- ( ph -> W e. LMod )
lspun0.x	|- ( ph -> X C_ V )

Assertion

Ref	Expression
lspun0	|- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` X ) )

Proof of Theorem lspun0

Step	Hyp	Ref	Expression
1		lspun0.w	. . 3 |- ( ph -> W e. LMod )
2		lspun0.x	. . 3 |- ( ph -> X C_ V )
3		lspun0.v	. . . . . 6 |- V = ( Base ` W )
4		lspun0.o	. . . . . 6 |- .0. = ( 0g ` W )
5	3, 4	lmod0vcl 14482	. . . . 5 |- ( W e. LMod -> .0. e. V )
6	1, 5	syl 14	. . . 4 |- ( ph -> .0. e. V )
7	6	snssd 3841	. . 3 |- ( ph -> { .0. } C_ V )
8		lspun0.n	. . . 4 |- N = ( LSpan ` W )
9	3, 8	lspun 14567	. . 3 |- ( ( W e. LMod /\ X C_ V /\ { .0. } C_ V ) -> ( N ` ( X u. { .0. } ) ) = ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) )
10	1, 2, 7, 9	syl3anc 1274	. 2 |- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) )
11	4, 8	lspsn0 14587	. . . . . . 7 |- ( W e. LMod -> ( N ` { .0. } ) = { .0. } )
12	1, 11	syl 14	. . . . . 6 |- ( ph -> ( N ` { .0. } ) = { .0. } )
13	12	uneq2d 3375	. . . . 5 |- ( ph -> ( ( N ` X ) u. ( N ` { .0. } ) ) = ( ( N ` X ) u. { .0. } ) )
14		eqid 2234	. . . . . . . . 9 |- ( LSubSp ` W ) = ( LSubSp ` W )
15	3, 14, 8	lspcl 14556	. . . . . . . 8 |- ( ( W e. LMod /\ X C_ V ) -> ( N ` X ) e. ( LSubSp ` W ) )
16	1, 2, 15	syl2anc 411	. . . . . . 7 |- ( ph -> ( N ` X ) e. ( LSubSp ` W ) )
17	4, 14	lss0ss 14536	. . . . . . 7 |- ( ( W e. LMod /\ ( N ` X ) e. ( LSubSp ` W ) ) -> { .0. } C_ ( N ` X ) )
18	1, 16, 17	syl2anc 411	. . . . . 6 |- ( ph -> { .0. } C_ ( N ` X ) )
19		ssequn2 3394	. . . . . 6 |- ( { .0. } C_ ( N ` X ) <-> ( ( N ` X ) u. { .0. } ) = ( N ` X ) )
20	18, 19	sylib 122	. . . . 5 |- ( ph -> ( ( N ` X ) u. { .0. } ) = ( N ` X ) )
21	13, 20	eqtrd 2267	. . . 4 |- ( ph -> ( ( N ` X ) u. ( N ` { .0. } ) ) = ( N ` X ) )
22	21	fveq2d 5676	. . 3 |- ( ph -> ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) = ( N ` ( N ` X ) ) )
23	3, 8	lspidm 14566	. . . 4 |- ( ( W e. LMod /\ X C_ V ) -> ( N ` ( N ` X ) ) = ( N ` X ) )
24	1, 2, 23	syl2anc 411	. . 3 |- ( ph -> ( N ` ( N ` X ) ) = ( N ` X ) )
25	22, 24	eqtrd 2267	. 2 |- ( ph -> ( N ` ( ( N ` X ) u. ( N ` { .0. } ) ) ) = ( N ` X ) )
26	10, 25	eqtrd 2267	1 |- ( ph -> ( N ` ( X u. { .0. } ) ) = ( N ` X ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   u. cun 3211   C_ wss 3213   {csn 3691   ` cfv 5354   Basecbs 13229   0gc0g 13486   LModclmod 14452   LSubSpclss 14517   LSpanclspn 14551

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 619  ax-in2 620  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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-sca 13323  df-vsca 13324  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-sbg 13735  df-mgp 14082  df-ur 14121  df-ring 14159  df-lmod 14454  df-lssm 14518  df-lsp 14552

This theorem is referenced by: (None)