Theorem lsateln0 34282

Description: A 1-dim subspace (atom) (of a left module or left vector space) contains a nonzero vector. (Contributed by NM, 2-Jan-2015.)

Hypotheses

Ref	Expression
lsateln0.z	|- .0. = ( 0g ` W )
lsateln0.a	|- A = (LSAtoms ` W )
lsateln0.w	|- ( ph -> W e. LMod )
lsateln0.u	|- ( ph -> U e. A )

Assertion

Ref	Expression
lsateln0	|- ( ph -> E. v e. U v =/= .0. )

Distinct variable groups:   v, U   v, W   v, .0.   ph, v

Allowed substitution hint:   A( v)

Proof of Theorem lsateln0

Step	Hyp	Ref	Expression
1		lsateln0.u	. . . 4 |- ( ph -> U e. A )
2		lsateln0.w	. . . . 5 |- ( ph -> W e. LMod )
3		eqid 2622	. . . . . 6 |- ( Base ` W ) = ( Base ` W )
4		eqid 2622	. . . . . 6 |- ( LSpan ` W ) = ( LSpan ` W )
5		lsateln0.z	. . . . . 6 |- .0. = ( 0g ` W )
6		lsateln0.a	. . . . . 6 |- A = (LSAtoms ` W )
7	3, 4, 5, 6	islsat 34278	. . . . 5 |- ( W e. LMod -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) ) )
8	2, 7	syl 17	. . . 4 |- ( ph -> ( U e. A <-> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) ) )
9	1, 8	mpbid 222	. . 3 |- ( ph -> E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) )
10		eldifi 3732	. . . . . 6 |- ( v e. ( ( Base ` W ) \ { .0. } ) -> v e. ( Base ` W ) )
11	3, 4	lspsnid 18993	. . . . . 6 |- ( ( W e. LMod /\ v e. ( Base ` W ) ) -> v e. ( ( LSpan ` W ) ` { v } ) )
12	2, 10, 11	syl2an 494	. . . . 5 |- ( ( ph /\ v e. ( ( Base ` W ) \ { .0. } ) ) -> v e. ( ( LSpan ` W ) ` { v } ) )
13		eleq2 2690	. . . . 5 |- ( U = ( ( LSpan ` W ) ` { v } ) -> ( v e. U <-> v e. ( ( LSpan ` W ) ` { v } ) ) )
14	12, 13	syl5ibrcom 237	. . . 4 |- ( ( ph /\ v e. ( ( Base ` W ) \ { .0. } ) ) -> ( U = ( ( LSpan ` W ) ` { v } ) -> v e. U ) )
15	14	reximdva 3017	. . 3 |- ( ph -> ( E. v e. ( ( Base ` W ) \ { .0. } ) U = ( ( LSpan ` W ) ` { v } ) -> E. v e. ( ( Base ` W ) \ { .0. } ) v e. U ) )
16	9, 15	mpd 15	. 2 |- ( ph -> E. v e. ( ( Base ` W ) \ { .0. } ) v e. U )
17		eldifsn 4317	. . . . . . 7 |- ( v e. ( ( Base ` W ) \ { .0. } ) <-> ( v e. ( Base ` W ) /\ v =/= .0. ) )
18	17	anbi1i 731	. . . . . 6 |- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) <-> ( ( v e. ( Base ` W ) /\ v =/= .0. ) /\ v e. U ) )
19		anass 681	. . . . . 6 |- ( ( ( v e. ( Base ` W ) /\ v =/= .0. ) /\ v e. U ) <-> ( v e. ( Base ` W ) /\ ( v =/= .0. /\ v e. U ) ) )
20	18, 19	bitri 264	. . . . 5 |- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) <-> ( v e. ( Base ` W ) /\ ( v =/= .0. /\ v e. U ) ) )
21	20	simprbi 480	. . . 4 |- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) -> ( v =/= .0. /\ v e. U ) )
22	21	ancomd 467	. . 3 |- ( ( v e. ( ( Base ` W ) \ { .0. } ) /\ v e. U ) -> ( v e. U /\ v =/= .0. ) )
23	22	reximi2 3010	. 2 |- ( E. v e. ( ( Base ` W ) \ { .0. } ) v e. U -> E. v e. U v =/= .0. )
24	16, 23	syl 17	1 |- ( ph -> E. v e. U v =/= .0. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   ` cfv 5888   Basecbs 15857   0gc0g 16100   LModclmod 18863   LSpanclspn 18971  LSAtomsclsa 34261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lsatoms 34263

This theorem is referenced by:  dvh1dim  36731  dochkr1  36767  dochkr1OLDN  36768  lcfrlem40  36871