Theorem lspsneu 19123

Description: Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)

Hypotheses

Ref	Expression
lspsneu.v	|- V = ( Base ` W )
lspsneu.s	|- S = (Scalar ` W )
lspsneu.k	|- K = ( Base ` S )
lspsneu.o	|- O = ( 0g ` S )
lspsneu.t	|- .x. = ( .s ` W )
lspsneu.z	|- .0. = ( 0g ` W )
lspsneu.n	|- N = ( LSpan ` W )
lspsneu.w	|- ( ph -> W e. LVec )
lspsneu.x	|- ( ph -> X e. V )
lspsneu.y	|- ( ph -> Y e. ( V \ { .0. } ) )

Assertion

Ref	Expression
lspsneu	|- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E! k e. ( K \ { O } ) X = ( k .x. Y ) ) )

Distinct variable groups:   k, K   k, O   .x. , k   k, X   k, Y

Allowed substitution hints:   ph( k)   S( k)   N( k)   V( k)   W( k)   .0. ( k)

Proof of Theorem lspsneu

Dummy variables i j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsneu.v	. . . . . . 7 |- V = ( Base ` W )
2		lspsneu.s	. . . . . . 7 |- S = (Scalar ` W )
3		lspsneu.k	. . . . . . 7 |- K = ( Base ` S )
4		lspsneu.o	. . . . . . 7 |- O = ( 0g ` S )
5		lspsneu.t	. . . . . . 7 |- .x. = ( .s ` W )
6		lspsneu.n	. . . . . . 7 |- N = ( LSpan ` W )
7		lspsneu.w	. . . . . . 7 |- ( ph -> W e. LVec )
8		lspsneu.x	. . . . . . 7 |- ( ph -> X e. V )
9		lspsneu.y	. . . . . . . 8 |- ( ph -> Y e. ( V \ { .0. } ) )
10	9	eldifad 3586	. . . . . . 7 |- ( ph -> Y e. V )
11	1, 2, 3, 4, 5, 6, 7, 8, 10	lspsneq 19122	. . . . . 6 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. j e. ( K \ { O } ) X = ( j .x. Y ) ) )
12	11	biimpd 219	. . . . 5 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> E. j e. ( K \ { O } ) X = ( j .x. Y ) ) )
13		eqtr2 2642	. . . . . . . . . 10 |- ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> ( j .x. Y ) = ( i .x. Y ) )
14	13	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> ( j .x. Y ) = ( i .x. Y ) )
15		lspsneu.z	. . . . . . . . . 10 |- .0. = ( 0g ` W )
16		simp1l 1085	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> ph )
17	16, 7	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> W e. LVec )
18		simp2l 1087	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> j e. ( K \ { O } ) )
19	18	eldifad 3586	. . . . . . . . . 10 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> j e. K )
20		simp2r 1088	. . . . . . . . . . 11 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> i e. ( K \ { O } ) )
21	20	eldifad 3586	. . . . . . . . . 10 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> i e. K )
22	16, 10	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> Y e. V )
23		eldifsni 4320	. . . . . . . . . . 11 |- ( Y e. ( V \ { .0. } ) -> Y =/= .0. )
24	16, 9, 23	3syl 18	. . . . . . . . . 10 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> Y =/= .0. )
25	1, 5, 2, 3, 15, 17, 19, 21, 22, 24	lvecvscan2 19112	. . . . . . . . 9 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> ( ( j .x. Y ) = ( i .x. Y ) <-> j = i ) )
26	14, 25	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) /\ ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) ) -> j = i )
27	26	3exp 1264	. . . . . . 7 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) -> ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> j = i ) ) )
28	27	ex 450	. . . . . 6 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> ( ( j e. ( K \ { O } ) /\ i e. ( K \ { O } ) ) -> ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> j = i ) ) ) )
29	28	ralrimdvv 2973	. . . . 5 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> A. j e. ( K \ { O } ) A. i e. ( K \ { O } ) ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> j = i ) ) )
30	12, 29	jcad 555	. . . 4 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> ( E. j e. ( K \ { O } ) X = ( j .x. Y ) /\ A. j e. ( K \ { O } ) A. i e. ( K \ { O } ) ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> j = i ) ) ) )
31		oveq1 6657	. . . . . 6 |- ( j = i -> ( j .x. Y ) = ( i .x. Y ) )
32	31	eqeq2d 2632	. . . . 5 |- ( j = i -> ( X = ( j .x. Y ) <-> X = ( i .x. Y ) ) )
33	32	reu4 3400	. . . 4 |- ( E! j e. ( K \ { O } ) X = ( j .x. Y ) <-> ( E. j e. ( K \ { O } ) X = ( j .x. Y ) /\ A. j e. ( K \ { O } ) A. i e. ( K \ { O } ) ( ( X = ( j .x. Y ) /\ X = ( i .x. Y ) ) -> j = i ) ) )
34	30, 33	syl6ibr 242	. . 3 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> E! j e. ( K \ { O } ) X = ( j .x. Y ) ) )
35		reurex 3160	. . . 4 |- ( E! j e. ( K \ { O } ) X = ( j .x. Y ) -> E. j e. ( K \ { O } ) X = ( j .x. Y ) )
36	35, 11	syl5ibr 236	. . 3 |- ( ph -> ( E! j e. ( K \ { O } ) X = ( j .x. Y ) -> ( N ` { X } ) = ( N ` { Y } ) ) )
37	34, 36	impbid 202	. 2 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E! j e. ( K \ { O } ) X = ( j .x. Y ) ) )
38		oveq1 6657	. . . 4 |- ( j = k -> ( j .x. Y ) = ( k .x. Y ) )
39	38	eqeq2d 2632	. . 3 |- ( j = k -> ( X = ( j .x. Y ) <-> X = ( k .x. Y ) ) )
40	39	cbvreuv 3173	. 2 |- ( E! j e. ( K \ { O } ) X = ( j .x. Y ) <-> E! k e. ( K \ { O } ) X = ( k .x. Y ) )
41	37, 40	syl6bb 276	1 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E! k e. ( K \ { O } ) X = ( k .x. Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   E!wreu 2914   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   LSpanclspn 18971   LVecclvec 19102

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  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-dvdsr 18641  df-unit 18642  df-invr 18672  df-drng 18749  df-lmod 18865  df-lss 18933  df-lsp 18972  df-lvec 19103

This theorem is referenced by:  hdmap14lem3  37162