Theorem lspsneq 19122

Description: Equal spans of singletons must have proportional vectors. See lspsnss2 19005 for comparable span version. TODO: can proof be shortened? (Contributed by NM, 21-Mar-2015.)

Hypotheses

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

Assertion

Ref	Expression
lspsneq	|- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. k e. ( K \ { .0. } ) X = ( k .x. Y ) ) )

Distinct variable groups:   k, K   .0. , k   .x. , k   k, X   k, Y

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

Proof of Theorem lspsneq

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lspsneq.w	. . . . . . . . . 10 |- ( ph -> W e. LVec )
2		lveclmod 19106	. . . . . . . . . 10 |- ( W e. LVec -> W e. LMod )
3	1, 2	syl 17	. . . . . . . . 9 |- ( ph -> W e. LMod )
4		lspsneq.s	. . . . . . . . . 10 |- S = (Scalar ` W )
5	4	lmodring 18871	. . . . . . . . 9 |- ( W e. LMod -> S e. Ring )
6		lspsneq.k	. . . . . . . . . 10 |- K = ( Base ` S )
7		eqid 2622	. . . . . . . . . 10 |- ( 1r ` S ) = ( 1r ` S )
8	6, 7	ringidcl 18568	. . . . . . . . 9 |- ( S e. Ring -> ( 1r ` S ) e. K )
9	3, 5, 8	3syl 18	. . . . . . . 8 |- ( ph -> ( 1r ` S ) e. K )
10	4	lvecdrng 19105	. . . . . . . . 9 |- ( W e. LVec -> S e. DivRing )
11		lspsneq.o	. . . . . . . . . 10 |- .0. = ( 0g ` S )
12	11, 7	drngunz 18762	. . . . . . . . 9 |- ( S e. DivRing -> ( 1r ` S ) =/= .0. )
13	1, 10, 12	3syl 18	. . . . . . . 8 |- ( ph -> ( 1r ` S ) =/= .0. )
14		eldifsn 4317	. . . . . . . 8 |- ( ( 1r ` S ) e. ( K \ { .0. } ) <-> ( ( 1r ` S ) e. K /\ ( 1r ` S ) =/= .0. ) )
15	9, 13, 14	sylanbrc 698	. . . . . . 7 |- ( ph -> ( 1r ` S ) e. ( K \ { .0. } ) )
16	15	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> ( 1r ` S ) e. ( K \ { .0. } ) )
17		lspsneq.v	. . . . . . . . . . 11 |- V = ( Base ` W )
18		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` W ) = ( 0g ` W )
19	17, 18	lmod0vcl 18892	. . . . . . . . . 10 |- ( W e. LMod -> ( 0g ` W ) e. V )
20	1, 2, 19	3syl 18	. . . . . . . . 9 |- ( ph -> ( 0g ` W ) e. V )
21		lspsneq.t	. . . . . . . . . 10 |- .x. = ( .s ` W )
22	17, 4, 21, 7	lmodvs1 18891	. . . . . . . . 9 |- ( ( W e. LMod /\ ( 0g ` W ) e. V ) -> ( ( 1r ` S ) .x. ( 0g ` W ) ) = ( 0g ` W ) )
23	3, 20, 22	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( 1r ` S ) .x. ( 0g ` W ) ) = ( 0g ` W ) )
24	23	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> ( ( 1r ` S ) .x. ( 0g ` W ) ) = ( 0g ` W ) )
25		oveq2 6658	. . . . . . . 8 |- ( Y = ( 0g ` W ) -> ( ( 1r ` S ) .x. Y ) = ( ( 1r ` S ) .x. ( 0g ` W ) ) )
26	25	adantl 482	. . . . . . 7 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> ( ( 1r ` S ) .x. Y ) = ( ( 1r ` S ) .x. ( 0g ` W ) ) )
27		lspsneq.n	. . . . . . . . 9 |- N = ( LSpan ` W )
28	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> W e. LMod )
29		lspsneq.x	. . . . . . . . . 10 |- ( ph -> X e. V )
30	29	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> X e. V )
31		lspsneq.y	. . . . . . . . . 10 |- ( ph -> Y e. V )
32	31	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> Y e. V )
33		simpr 477	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( N ` { X } ) = ( N ` { Y } ) )
34	17, 18, 27, 28, 30, 32, 33	lspsneq0b 19013	. . . . . . . 8 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( X = ( 0g ` W ) <-> Y = ( 0g ` W ) ) )
35	34	biimpar 502	. . . . . . 7 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> X = ( 0g ` W ) )
36	24, 26, 35	3eqtr4rd 2667	. . . . . 6 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> X = ( ( 1r ` S ) .x. Y ) )
37		oveq1 6657	. . . . . . . 8 |- ( j = ( 1r ` S ) -> ( j .x. Y ) = ( ( 1r ` S ) .x. Y ) )
38	37	eqeq2d 2632	. . . . . . 7 |- ( j = ( 1r ` S ) -> ( X = ( j .x. Y ) <-> X = ( ( 1r ` S ) .x. Y ) ) )
39	38	rspcev 3309	. . . . . 6 |- ( ( ( 1r ` S ) e. ( K \ { .0. } ) /\ X = ( ( 1r ` S ) .x. Y ) ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) )
40	16, 36, 39	syl2anc 693	. . . . 5 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y = ( 0g ` W ) ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) )
41		eqimss 3657	. . . . . . . . . 10 |- ( ( N ` { X } ) = ( N ` { Y } ) -> ( N ` { X } ) C_ ( N ` { Y } ) )
42	41	adantl 482	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( N ` { X } ) C_ ( N ` { Y } ) )
43		eqid 2622	. . . . . . . . . 10 |- ( LSubSp ` W ) = ( LSubSp ` W )
44	17, 43, 27	lspsncl 18977	. . . . . . . . . . . 12 |- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
45	3, 31, 44	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
46	45	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
47	17, 43, 27, 28, 46, 30	lspsnel5 18995	. . . . . . . . 9 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( X e. ( N ` { Y } ) <-> ( N ` { X } ) C_ ( N ` { Y } ) ) )
48	42, 47	mpbird 247	. . . . . . . 8 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> X e. ( N ` { Y } ) )
49	4, 6, 17, 21, 27	lspsnel 19003	. . . . . . . . 9 |- ( ( W e. LMod /\ Y e. V ) -> ( X e. ( N ` { Y } ) <-> E. j e. K X = ( j .x. Y ) ) )
50	28, 32, 49	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( X e. ( N ` { Y } ) <-> E. j e. K X = ( j .x. Y ) ) )
51	48, 50	mpbid 222	. . . . . . 7 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> E. j e. K X = ( j .x. Y ) )
52	51	adantr 481	. . . . . 6 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) -> E. j e. K X = ( j .x. Y ) )
53		simprl 794	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> j e. K )
54		simpr 477	. . . . . . . . . . . . . 14 |- ( ( j e. K /\ X = ( j .x. Y ) ) -> X = ( j .x. Y ) )
55	54	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> X = ( j .x. Y ) )
56	34	biimpd 219	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( X = ( 0g ` W ) -> Y = ( 0g ` W ) ) )
57	56	necon3d 2815	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> ( Y =/= ( 0g ` W ) -> X =/= ( 0g ` W ) ) )
58	57	imp 445	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) -> X =/= ( 0g ` W ) )
59	58	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> X =/= ( 0g ` W ) )
60	55, 59	eqnetrrd 2862	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> ( j .x. Y ) =/= ( 0g ` W ) )
61	1	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> W e. LVec )
62	61	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> W e. LVec )
63	32	ad2antrr 762	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> Y e. V )
64	17, 21, 4, 6, 11, 18, 62, 53, 63	lvecvsn0 19109	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> ( ( j .x. Y ) =/= ( 0g ` W ) <-> ( j =/= .0. /\ Y =/= ( 0g ` W ) ) ) )
65	60, 64	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> ( j =/= .0. /\ Y =/= ( 0g ` W ) ) )
66	65	simpld 475	. . . . . . . . . 10 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> j =/= .0. )
67		eldifsn 4317	. . . . . . . . . 10 |- ( j e. ( K \ { .0. } ) <-> ( j e. K /\ j =/= .0. ) )
68	53, 66, 67	sylanbrc 698	. . . . . . . . 9 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> j e. ( K \ { .0. } ) )
69	68, 55	jca 554	. . . . . . . 8 |- ( ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) /\ ( j e. K /\ X = ( j .x. Y ) ) ) -> ( j e. ( K \ { .0. } ) /\ X = ( j .x. Y ) ) )
70	69	ex 450	. . . . . . 7 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) -> ( ( j e. K /\ X = ( j .x. Y ) ) -> ( j e. ( K \ { .0. } ) /\ X = ( j .x. Y ) ) ) )
71	70	reximdv2 3014	. . . . . 6 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) -> ( E. j e. K X = ( j .x. Y ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) ) )
72	52, 71	mpd 15	. . . . 5 |- ( ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) /\ Y =/= ( 0g ` W ) ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) )
73	40, 72	pm2.61dane 2881	. . . 4 |- ( ( ph /\ ( N ` { X } ) = ( N ` { Y } ) ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) )
74	73	ex 450	. . 3 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) -> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) ) )
75	1	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. ( K \ { .0. } ) ) -> W e. LVec )
76		eldifi 3732	. . . . . . . 8 |- ( j e. ( K \ { .0. } ) -> j e. K )
77	76	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. ( K \ { .0. } ) ) -> j e. K )
78		eldifsni 4320	. . . . . . . 8 |- ( j e. ( K \ { .0. } ) -> j =/= .0. )
79	78	adantl 482	. . . . . . 7 |- ( ( ph /\ j e. ( K \ { .0. } ) ) -> j =/= .0. )
80	31	adantr 481	. . . . . . 7 |- ( ( ph /\ j e. ( K \ { .0. } ) ) -> Y e. V )
81	17, 4, 21, 6, 11, 27	lspsnvs 19114	. . . . . . 7 |- ( ( W e. LVec /\ ( j e. K /\ j =/= .0. ) /\ Y e. V ) -> ( N ` { ( j .x. Y ) } ) = ( N ` { Y } ) )
82	75, 77, 79, 80, 81	syl121anc 1331	. . . . . 6 |- ( ( ph /\ j e. ( K \ { .0. } ) ) -> ( N ` { ( j .x. Y ) } ) = ( N ` { Y } ) )
83	82	ex 450	. . . . 5 |- ( ph -> ( j e. ( K \ { .0. } ) -> ( N ` { ( j .x. Y ) } ) = ( N ` { Y } ) ) )
84		sneq 4187	. . . . . . . 8 |- ( X = ( j .x. Y ) -> { X } = { ( j .x. Y ) } )
85	84	fveq2d 6195	. . . . . . 7 |- ( X = ( j .x. Y ) -> ( N ` { X } ) = ( N ` { ( j .x. Y ) } ) )
86	85	eqeq1d 2624	. . . . . 6 |- ( X = ( j .x. Y ) -> ( ( N ` { X } ) = ( N ` { Y } ) <-> ( N ` { ( j .x. Y ) } ) = ( N ` { Y } ) ) )
87	86	biimprcd 240	. . . . 5 |- ( ( N ` { ( j .x. Y ) } ) = ( N ` { Y } ) -> ( X = ( j .x. Y ) -> ( N ` { X } ) = ( N ` { Y } ) ) )
88	83, 87	syl6 35	. . . 4 |- ( ph -> ( j e. ( K \ { .0. } ) -> ( X = ( j .x. Y ) -> ( N ` { X } ) = ( N ` { Y } ) ) ) )
89	88	rexlimdv 3030	. . 3 |- ( ph -> ( E. j e. ( K \ { .0. } ) X = ( j .x. Y ) -> ( N ` { X } ) = ( N ` { Y } ) ) )
90	74, 89	impbid 202	. 2 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. j e. ( K \ { .0. } ) X = ( j .x. Y ) ) )
91		oveq1 6657	. . . 4 |- ( j = k -> ( j .x. Y ) = ( k .x. Y ) )
92	91	eqeq2d 2632	. . 3 |- ( j = k -> ( X = ( j .x. Y ) <-> X = ( k .x. Y ) ) )
93	92	cbvrexv 3172	. 2 |- ( E. j e. ( K \ { .0. } ) X = ( j .x. Y ) <-> E. k e. ( K \ { .0. } ) X = ( k .x. Y ) )
94	90, 93	syl6bb 276	1 |- ( ph -> ( ( N ` { X } ) = ( N ` { Y } ) <-> E. k e. ( K \ { .0. } ) X = ( k .x. Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   1rcur 18501   Ringcrg 18547   DivRingcdr 18747   LModclmod 18863   LSubSpclss 18932   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:  lspsneu  19123  mapdpglem26  36987  mapdpglem27  36988  hdmap14lem2a  37159  hdmap14lem2N  37161