Theorem lspexch 19129

Description: Exchange property for span of a pair. TODO: see if a version with Y,Z and X,Z reversed will shorten proofs (analogous to lspexchn1 19130 vs. lspexchn2 19131); look for lspexch 19129 and prcom 4267 in same proof. TODO: would a hypothesis of -. X e. ( N ` { Z } ) instead of ( N ` { X } ) =/= ( N { Z } ) ` be better overall? This would be shorter and also satisfy the X =/= .0. condition. Here and also lspindp* and all proofs affected by them (all in NM's mathbox); there are 58 hypotheses with the =/= pattern as of 24-May-2015. (Contributed by NM, 11-Apr-2015.)

Hypotheses

Ref	Expression
lspexch.v	|- V = ( Base ` W )
lspexch.o	|- .0. = ( 0g ` W )
lspexch.n	|- N = ( LSpan ` W )
lspexch.w	|- ( ph -> W e. LVec )
lspexch.x	|- ( ph -> X e. ( V \ { .0. } ) )
lspexch.y	|- ( ph -> Y e. V )
lspexch.z	|- ( ph -> Z e. V )
lspexch.q	|- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
lspexch.e	|- ( ph -> X e. ( N ` { Y , Z } ) )

Assertion

Ref	Expression
lspexch	|- ( ph -> Y e. ( N ` { X , Z } ) )

Proof of Theorem lspexch

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

Step	Hyp	Ref	Expression
1		lspexch.e	. . 3 |- ( ph -> X e. ( N ` { Y , Z } ) )
2		lspexch.v	. . . 4 |- V = ( Base ` W )
3		eqid 2622	. . . 4 |- ( +g ` W ) = ( +g ` W )
4		eqid 2622	. . . 4 |- (Scalar ` W ) = (Scalar ` W )
5		eqid 2622	. . . 4 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
6		eqid 2622	. . . 4 |- ( .s ` W ) = ( .s ` W )
7		lspexch.n	. . . 4 |- N = ( LSpan ` W )
8		lspexch.w	. . . . 5 |- ( ph -> W e. LVec )
9		lveclmod 19106	. . . . 5 |- ( W e. LVec -> W e. LMod )
10	8, 9	syl 17	. . . 4 |- ( ph -> W e. LMod )
11		lspexch.y	. . . 4 |- ( ph -> Y e. V )
12		lspexch.z	. . . 4 |- ( ph -> Z e. V )
13	2, 3, 4, 5, 6, 7, 10, 11, 12	lspprel 19094	. . 3 |- ( ph -> ( X e. ( N ` { Y , Z } ) <-> E. j e. ( Base ` (Scalar ` W ) ) E. k e. ( Base ` (Scalar ` W ) ) X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) )
14	1, 13	mpbid 222	. 2 |- ( ph -> E. j e. ( Base ` (Scalar ` W ) ) E. k e. ( Base ` (Scalar ` W ) ) X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) )
15		eqid 2622	. . . . . . . 8 |- ( -g ` W ) = ( -g ` W )
16		eqid 2622	. . . . . . . 8 |- ( invg ` (Scalar ` W ) ) = ( invg ` (Scalar ` W ) )
17	8	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> W e. LVec )
18	17, 9	syl 17	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> W e. LMod )
19		simp2r 1088	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> k e. ( Base ` (Scalar ` W ) ) )
20		lspexch.x	. . . . . . . . . 10 |- ( ph -> X e. ( V \ { .0. } ) )
21	20	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> X e. ( V \ { .0. } ) )
22	21	eldifad 3586	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> X e. V )
23	12	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> Z e. V )
24	2, 3, 15, 6, 4, 5, 16, 18, 19, 22, 23	lmodsubvs 18919	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( X ( -g ` W ) ( k ( .s ` W ) Z ) ) = ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) )
25		simp3 1063	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) )
26	25	eqcomd 2628	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = X )
27	10	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> W e. LMod )
28		lmodgrp 18870	. . . . . . . . . 10 |- ( W e. LMod -> W e. Grp )
29	27, 28	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> W e. Grp )
30	2, 4, 6, 5	lmodvscl 18880	. . . . . . . . . 10 |- ( ( W e. LMod /\ k e. ( Base ` (Scalar ` W ) ) /\ Z e. V ) -> ( k ( .s ` W ) Z ) e. V )
31	18, 19, 23, 30	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( k ( .s ` W ) Z ) e. V )
32		simp2l 1087	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> j e. ( Base ` (Scalar ` W ) ) )
33	11	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> Y e. V )
34	2, 4, 6, 5	lmodvscl 18880	. . . . . . . . . 10 |- ( ( W e. LMod /\ j e. ( Base ` (Scalar ` W ) ) /\ Y e. V ) -> ( j ( .s ` W ) Y ) e. V )
35	18, 32, 33, 34	syl3anc 1326	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( j ( .s ` W ) Y ) e. V )
36	2, 3, 15	grpsubadd 17503	. . . . . . . . 9 |- ( ( W e. Grp /\ ( X e. V /\ ( k ( .s ` W ) Z ) e. V /\ ( j ( .s ` W ) Y ) e. V ) ) -> ( ( X ( -g ` W ) ( k ( .s ` W ) Z ) ) = ( j ( .s ` W ) Y ) <-> ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = X ) )
37	29, 22, 31, 35, 36	syl13anc 1328	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( X ( -g ` W ) ( k ( .s ` W ) Z ) ) = ( j ( .s ` W ) Y ) <-> ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = X ) )
38	26, 37	mpbird 247	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( X ( -g ` W ) ( k ( .s ` W ) Z ) ) = ( j ( .s ` W ) Y ) )
39	24, 38	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) = ( j ( .s ` W ) Y ) )
40		eqid 2622	. . . . . . 7 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
41		eqid 2622	. . . . . . 7 |- ( invr ` (Scalar ` W ) ) = ( invr ` (Scalar ` W ) )
42		lspexch.q	. . . . . . . . . 10 |- ( ph -> ( N ` { X } ) =/= ( N ` { Z } ) )
43	42	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( N ` { X } ) =/= ( N ` { Z } ) )
44		lspexch.o	. . . . . . . . . . . 12 |- .0. = ( 0g ` W )
45	17	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> W e. LVec )
46	23	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> Z e. V )
47	25	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) )
48		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( j = ( 0g ` (Scalar ` W ) ) -> ( j ( .s ` W ) Y ) = ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) )
49	48	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( j = ( 0g ` (Scalar ` W ) ) -> ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) )
50	2, 4, 6, 40, 44	lmod0vs 18896	. . . . . . . . . . . . . . . . . 18 |- ( ( W e. LMod /\ Y e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) = .0. )
51	18, 33, 50	syl2anc 693	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) = .0. )
52	51	oveq1d 6665	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( .0. ( +g ` W ) ( k ( .s ` W ) Z ) ) )
53	2, 3, 44	lmod0vlid 18893	. . . . . . . . . . . . . . . . 17 |- ( ( W e. LMod /\ ( k ( .s ` W ) Z ) e. V ) -> ( .0. ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( k ( .s ` W ) Z ) )
54	18, 31, 53	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( .0. ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( k ( .s ` W ) Z ) )
55	52, 54	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( k ( .s ` W ) Z ) )
56	49, 55	sylan9eqr 2678	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) = ( k ( .s ` W ) Z ) )
57	47, 56	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> X = ( k ( .s ` W ) Z ) )
58	2, 6, 4, 5, 7, 18, 19, 23	lspsneli 19001	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( k ( .s ` W ) Z ) e. ( N ` { Z } ) )
59	58	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> ( k ( .s ` W ) Z ) e. ( N ` { Z } ) )
60	57, 59	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> X e. ( N ` { Z } ) )
61		eldifsni 4320	. . . . . . . . . . . . . 14 |- ( X e. ( V \ { .0. } ) -> X =/= .0. )
62	21, 61	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> X =/= .0. )
63	62	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> X =/= .0. )
64	2, 44, 7, 45, 46, 60, 63	lspsneleq 19115	. . . . . . . . . . 11 |- ( ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) /\ j = ( 0g ` (Scalar ` W ) ) ) -> ( N ` { X } ) = ( N ` { Z } ) )
65	64	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( j = ( 0g ` (Scalar ` W ) ) -> ( N ` { X } ) = ( N ` { Z } ) ) )
66	65	necon3d 2815	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( N ` { X } ) =/= ( N ` { Z } ) -> j =/= ( 0g ` (Scalar ` W ) ) ) )
67	43, 66	mpd 15	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> j =/= ( 0g ` (Scalar ` W ) ) )
68		eldifsn 4317	. . . . . . . 8 |- ( j e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) <-> ( j e. ( Base ` (Scalar ` W ) ) /\ j =/= ( 0g ` (Scalar ` W ) ) ) )
69	32, 67, 68	sylanbrc 698	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> j e. ( ( Base ` (Scalar ` W ) ) \ { ( 0g ` (Scalar ` W ) ) } ) )
70	4	lmodfgrp 18872	. . . . . . . . . . 11 |- ( W e. LMod -> (Scalar ` W ) e. Grp )
71	27, 70	syl 17	. . . . . . . . . 10 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> (Scalar ` W ) e. Grp )
72	5, 16	grpinvcl 17467	. . . . . . . . . 10 |- ( ( (Scalar ` W ) e. Grp /\ k e. ( Base ` (Scalar ` W ) ) ) -> ( ( invg ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) )
73	71, 19, 72	syl2anc 693	. . . . . . . . 9 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( invg ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) )
74	2, 4, 6, 5	lmodvscl 18880	. . . . . . . . 9 |- ( ( W e. LMod /\ ( ( invg ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) /\ Z e. V ) -> ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) e. V )
75	18, 73, 23, 74	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) e. V )
76	2, 3	lmodvacl 18877	. . . . . . . 8 |- ( ( W e. LMod /\ X e. V /\ ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) e. V ) -> ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) e. V )
77	18, 22, 75, 76	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) e. V )
78	2, 6, 4, 5, 40, 41, 17, 69, 77, 33	lvecinv 19113	. . . . . 6 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) = ( j ( .s ` W ) Y ) <-> Y = ( ( ( invr ` (Scalar ` W ) ) ` j ) ( .s ` W ) ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) ) ) )
79	39, 78	mpbid 222	. . . . 5 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> Y = ( ( ( invr ` (Scalar ` W ) ) ` j ) ( .s ` W ) ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) ) )
80		eqid 2622	. . . . . . 7 |- ( LSubSp ` W ) = ( LSubSp ` W )
81	2, 80, 7, 18, 22, 23	lspprcl 18978	. . . . . 6 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( N ` { X , Z } ) e. ( LSubSp ` W ) )
82	4	lvecdrng 19105	. . . . . . . 8 |- ( W e. LVec -> (Scalar ` W ) e. DivRing )
83	17, 82	syl 17	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> (Scalar ` W ) e. DivRing )
84	5, 40, 41	drnginvrcl 18764	. . . . . . 7 |- ( ( (Scalar ` W ) e. DivRing /\ j e. ( Base ` (Scalar ` W ) ) /\ j =/= ( 0g ` (Scalar ` W ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` j ) e. ( Base ` (Scalar ` W ) ) )
85	83, 32, 67, 84	syl3anc 1326	. . . . . 6 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` j ) e. ( Base ` (Scalar ` W ) ) )
86		eqid 2622	. . . . . . . . . 10 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
87	2, 4, 6, 86	lmodvs1 18891	. . . . . . . . 9 |- ( ( W e. LMod /\ X e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) = X )
88	18, 22, 87	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) = X )
89	88	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) = ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) )
90	4	lmodring 18871	. . . . . . . . 9 |- ( W e. LMod -> (Scalar ` W ) e. Ring )
91	5, 86	ringidcl 18568	. . . . . . . . 9 |- ( (Scalar ` W ) e. Ring -> ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
92	18, 90, 91	3syl 18	. . . . . . . 8 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( 1r ` (Scalar ` W ) ) e. ( Base ` (Scalar ` W ) ) )
93	2, 3, 6, 4, 5, 7, 18, 92, 73, 22, 23	lsppreli 19090	. . . . . . 7 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) X ) ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) e. ( N ` { X , Z } ) )
94	89, 93	eqeltrrd 2702	. . . . . 6 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) e. ( N ` { X , Z } ) )
95	4, 6, 5, 80	lssvscl 18955	. . . . . 6 |- ( ( ( W e. LMod /\ ( N ` { X , Z } ) e. ( LSubSp ` W ) ) /\ ( ( ( invr ` (Scalar ` W ) ) ` j ) e. ( Base ` (Scalar ` W ) ) /\ ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) e. ( N ` { X , Z } ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` j ) ( .s ` W ) ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) ) e. ( N ` { X , Z } ) )
96	18, 81, 85, 94, 95	syl22anc 1327	. . . . 5 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` j ) ( .s ` W ) ( X ( +g ` W ) ( ( ( invg ` (Scalar ` W ) ) ` k ) ( .s ` W ) Z ) ) ) e. ( N ` { X , Z } ) )
97	79, 96	eqeltrd 2701	. . . 4 |- ( ( ph /\ ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) ) -> Y e. ( N ` { X , Z } ) )
98	97	3exp 1264	. . 3 |- ( ph -> ( ( j e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) ) -> ( X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) -> Y e. ( N ` { X , Z } ) ) ) )
99	98	rexlimdvv 3037	. 2 |- ( ph -> ( E. j e. ( Base ` (Scalar ` W ) ) E. k e. ( Base ` (Scalar ` W ) ) X = ( ( j ( .s ` W ) Y ) ( +g ` W ) ( k ( .s ` W ) Z ) ) -> Y e. ( N ` { X , Z } ) ) )
100	14, 99	mpd 15	1 |- ( ph -> Y e. ( N ` { X , Z } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   invgcminusg 17423   -gcsg 17424   1rcur 18501   Ringcrg 18547   invrcinvr 18671   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-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-cmn 18195  df-abl 18196  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:  lspexchn1  19130  lspindp1  19133  mapdh8ab  37066  mapdh8ad  37068  mapdh8b  37069  mapdh8c  37070  mapdh8e  37073