Theorem lspfixed 19128

Description: Show membership in the span of the sum of two vectors, one of which ( Y) is fixed in advance. (Contributed by NM, 27-May-2015.)

Hypotheses

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

Assertion

Ref	Expression
lspfixed	|- ( ph -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) )

Distinct variable groups:   z, N   z, .0.   z, .+   z, W   z, X   z, Y   z, Z

Allowed substitution hints:   ph( z)   V( z)

Proof of Theorem lspfixed

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

Step	Hyp	Ref	Expression
1		lspfixed.g	. . 3 |- ( ph -> X e. ( N ` { Y , Z } ) )
2		lspfixed.v	. . . 4 |- V = ( Base ` W )
3		lspfixed.p	. . . 4 |- .+ = ( +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		lspfixed.n	. . . 4 |- N = ( LSpan ` W )
8		lspfixed.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		lspfixed.y	. . . 4 |- ( ph -> Y e. V )
12		lspfixed.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. k e. ( Base ` (Scalar ` W ) ) E. l e. ( Base ` (Scalar ` W ) ) X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) )
14	1, 13	mpbid 222	. 2 |- ( ph -> E. k e. ( Base ` (Scalar ` W ) ) E. l e. ( Base ` (Scalar ` W ) ) X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) )
15	10	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> W e. LMod )
16		eqid 2622	. . . . . . . . . 10 |- ( LSubSp ` W ) = ( LSubSp ` W )
17	2, 16, 7	lspsncl 18977	. . . . . . . . 9 |- ( ( W e. LMod /\ Z e. V ) -> ( N ` { Z } ) e. ( LSubSp ` W ) )
18	10, 12, 17	syl2anc 693	. . . . . . . 8 |- ( ph -> ( N ` { Z } ) e. ( LSubSp ` W ) )
19	18	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { Z } ) e. ( LSubSp ` W ) )
20	8	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> W e. LVec )
21	4	lvecdrng 19105	. . . . . . . . 9 |- ( W e. LVec -> (Scalar ` W ) e. DivRing )
22	20, 21	syl 17	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> (Scalar ` W ) e. DivRing )
23		simp2l 1087	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> k e. ( Base ` (Scalar ` W ) ) )
24		lspfixed.f	. . . . . . . . . 10 |- ( ph -> -. X e. ( N ` { Z } ) )
25	24	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> -. X e. ( N ` { Z } ) )
26		simpl3 1066	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) )
27		simpr 477	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> k = ( 0g ` (Scalar ` W ) ) )
28	27	oveq1d 6665	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( k ( .s ` W ) Y ) = ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) )
29		simpl1 1064	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ph )
30	29, 10	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> W e. LMod )
31	29, 11	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> Y e. V )
32		eqid 2622	. . . . . . . . . . . . . . . . 17 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
33		lspfixed.o	. . . . . . . . . . . . . . . . 17 |- .0. = ( 0g ` W )
34	2, 4, 6, 32, 33	lmod0vs 18896	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ Y e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) = .0. )
35	30, 31, 34	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Y ) = .0. )
36	28, 35	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( k ( .s ` W ) Y ) = .0. )
37	36	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) = ( .0. .+ ( l ( .s ` W ) Z ) ) )
38		simp2r 1088	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> l e. ( Base ` (Scalar ` W ) ) )
39	12	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> Z e. V )
40	2, 4, 6, 5	lmodvscl 18880	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ l e. ( Base ` (Scalar ` W ) ) /\ Z e. V ) -> ( l ( .s ` W ) Z ) e. V )
41	15, 38, 39, 40	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( l ( .s ` W ) Z ) e. V )
42	41	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( l ( .s ` W ) Z ) e. V )
43	2, 3, 33	lmod0vlid 18893	. . . . . . . . . . . . . 14 |- ( ( W e. LMod /\ ( l ( .s ` W ) Z ) e. V ) -> ( .0. .+ ( l ( .s ` W ) Z ) ) = ( l ( .s ` W ) Z ) )
44	30, 42, 43	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( .0. .+ ( l ( .s ` W ) Z ) ) = ( l ( .s ` W ) Z ) )
45	26, 37, 44	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> X = ( l ( .s ` W ) Z ) )
46	29, 18	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( N ` { Z } ) e. ( LSubSp ` W ) )
47		simpl2r 1115	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> l e. ( Base ` (Scalar ` W ) ) )
48	2, 7	lspsnid 18993	. . . . . . . . . . . . . . 15 |- ( ( W e. LMod /\ Z e. V ) -> Z e. ( N ` { Z } ) )
49	10, 12, 48	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> Z e. ( N ` { Z } ) )
50	29, 49	syl 17	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> Z e. ( N ` { Z } ) )
51	4, 6, 5, 16	lssvscl 18955	. . . . . . . . . . . . 13 |- ( ( ( W e. LMod /\ ( N ` { Z } ) e. ( LSubSp ` W ) ) /\ ( l e. ( Base ` (Scalar ` W ) ) /\ Z e. ( N ` { Z } ) ) ) -> ( l ( .s ` W ) Z ) e. ( N ` { Z } ) )
52	30, 46, 47, 50, 51	syl22anc 1327	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> ( l ( .s ` W ) Z ) e. ( N ` { Z } ) )
53	45, 52	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ k = ( 0g ` (Scalar ` W ) ) ) -> X e. ( N ` { Z } ) )
54	53	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( k = ( 0g ` (Scalar ` W ) ) -> X e. ( N ` { Z } ) ) )
55	54	necon3bd 2808	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( -. X e. ( N ` { Z } ) -> k =/= ( 0g ` (Scalar ` W ) ) ) )
56	25, 55	mpd 15	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> k =/= ( 0g ` (Scalar ` W ) ) )
57		eqid 2622	. . . . . . . . 9 |- ( invr ` (Scalar ` W ) ) = ( invr ` (Scalar ` W ) )
58	5, 32, 57	drnginvrcl 18764	. . . . . . . 8 |- ( ( (Scalar ` W ) e. DivRing /\ k e. ( Base ` (Scalar ` W ) ) /\ k =/= ( 0g ` (Scalar ` W ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) )
59	22, 23, 56, 58	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) )
60	49	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> Z e. ( N ` { Z } ) )
61	15, 19, 38, 60, 51	syl22anc 1327	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( l ( .s ` W ) Z ) e. ( N ` { Z } ) )
62	4, 6, 5, 16	lssvscl 18955	. . . . . . 7 |- ( ( ( W e. LMod /\ ( N ` { Z } ) e. ( LSubSp ` W ) ) /\ ( ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) /\ ( l ( .s ` W ) Z ) e. ( N ` { Z } ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( N ` { Z } ) )
63	15, 19, 59, 61, 62	syl22anc 1327	. . . . . 6 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( N ` { Z } ) )
64	5, 32, 57	drnginvrn0 18765	. . . . . . . 8 |- ( ( (Scalar ` W ) e. DivRing /\ k e. ( Base ` (Scalar ` W ) ) /\ k =/= ( 0g ` (Scalar ` W ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` k ) =/= ( 0g ` (Scalar ` W ) ) )
65	22, 23, 56, 64	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( invr ` (Scalar ` W ) ) ` k ) =/= ( 0g ` (Scalar ` W ) ) )
66		lspfixed.e	. . . . . . . . . 10 |- ( ph -> -. X e. ( N ` { Y } ) )
67	66	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> -. X e. ( N ` { Y } ) )
68		simpl3 1066	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) )
69		oveq1 6657	. . . . . . . . . . . . . . 15 |- ( l = ( 0g ` (Scalar ` W ) ) -> ( l ( .s ` W ) Z ) = ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Z ) )
70	2, 4, 6, 32, 33	lmod0vs 18896	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ Z e. V ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Z ) = .0. )
71	15, 39, 70	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( 0g ` (Scalar ` W ) ) ( .s ` W ) Z ) = .0. )
72	69, 71	sylan9eqr 2678	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> ( l ( .s ` W ) Z ) = .0. )
73	72	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) = ( ( k ( .s ` W ) Y ) .+ .0. ) )
74	11	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> Y e. V )
75	2, 4, 6, 5	lmodvscl 18880	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ k e. ( Base ` (Scalar ` W ) ) /\ Y e. V ) -> ( k ( .s ` W ) Y ) e. V )
76	15, 23, 74, 75	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( k ( .s ` W ) Y ) e. V )
77	2, 3, 33	lmod0vrid 18894	. . . . . . . . . . . . . . 15 |- ( ( W e. LMod /\ ( k ( .s ` W ) Y ) e. V ) -> ( ( k ( .s ` W ) Y ) .+ .0. ) = ( k ( .s ` W ) Y ) )
78	15, 76, 77	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( k ( .s ` W ) Y ) .+ .0. ) = ( k ( .s ` W ) Y ) )
79	78	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> ( ( k ( .s ` W ) Y ) .+ .0. ) = ( k ( .s ` W ) Y ) )
80	68, 73, 79	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> X = ( k ( .s ` W ) Y ) )
81	2, 16, 7	lspsncl 18977	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
82	10, 11, 81	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> ( N ` { Y } ) e. ( LSubSp ` W ) )
83	82	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { Y } ) e. ( LSubSp ` W ) )
84	2, 7	lspsnid 18993	. . . . . . . . . . . . . . . 16 |- ( ( W e. LMod /\ Y e. V ) -> Y e. ( N ` { Y } ) )
85	10, 11, 84	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> Y e. ( N ` { Y } ) )
86	85	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> Y e. ( N ` { Y } ) )
87	4, 6, 5, 16	lssvscl 18955	. . . . . . . . . . . . . 14 |- ( ( ( W e. LMod /\ ( N ` { Y } ) e. ( LSubSp ` W ) ) /\ ( k e. ( Base ` (Scalar ` W ) ) /\ Y e. ( N ` { Y } ) ) ) -> ( k ( .s ` W ) Y ) e. ( N ` { Y } ) )
88	15, 83, 23, 86, 87	syl22anc 1327	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( k ( .s ` W ) Y ) e. ( N ` { Y } ) )
89	88	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> ( k ( .s ` W ) Y ) e. ( N ` { Y } ) )
90	80, 89	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ l = ( 0g ` (Scalar ` W ) ) ) -> X e. ( N ` { Y } ) )
91	90	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( l = ( 0g ` (Scalar ` W ) ) -> X e. ( N ` { Y } ) ) )
92	91	necon3bd 2808	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( -. X e. ( N ` { Y } ) -> l =/= ( 0g ` (Scalar ` W ) ) ) )
93	67, 92	mpd 15	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> l =/= ( 0g ` (Scalar ` W ) ) )
94		simpl1 1064	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ Z = .0. ) -> ph )
95	94, 1	syl 17	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ Z = .0. ) -> X e. ( N ` { Y , Z } ) )
96		preq2 4269	. . . . . . . . . . . . . 14 |- ( Z = .0. -> { Y , Z } = { Y , .0. } )
97	96	fveq2d 6195	. . . . . . . . . . . . 13 |- ( Z = .0. -> ( N ` { Y , Z } ) = ( N ` { Y , .0. } ) )
98	2, 33, 7, 15, 74	lsppr0 19092	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { Y , .0. } ) = ( N ` { Y } ) )
99	97, 98	sylan9eqr 2678	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ Z = .0. ) -> ( N ` { Y , Z } ) = ( N ` { Y } ) )
100	95, 99	eleqtrd 2703	. . . . . . . . . . 11 |- ( ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) /\ Z = .0. ) -> X e. ( N ` { Y } ) )
101	100	ex 450	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( Z = .0. -> X e. ( N ` { Y } ) ) )
102	101	necon3bd 2808	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( -. X e. ( N ` { Y } ) -> Z =/= .0. ) )
103	67, 102	mpd 15	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> Z =/= .0. )
104	2, 6, 4, 5, 32, 33, 20, 38, 39	lvecvsn0 19109	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( l ( .s ` W ) Z ) =/= .0. <-> ( l =/= ( 0g ` (Scalar ` W ) ) /\ Z =/= .0. ) ) )
105	93, 103, 104	mpbir2and 957	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( l ( .s ` W ) Z ) =/= .0. )
106	2, 6, 4, 5, 32, 33, 20, 59, 41	lvecvsn0 19109	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) =/= .0. <-> ( ( ( invr ` (Scalar ` W ) ) ` k ) =/= ( 0g ` (Scalar ` W ) ) /\ ( l ( .s ` W ) Z ) =/= .0. ) ) )
107	65, 105, 106	mpbir2and 957	. . . . . 6 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) =/= .0. )
108		eldifsn 4317	. . . . . 6 |- ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( ( N ` { Z } ) \ { .0. } ) <-> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( N ` { Z } ) /\ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) =/= .0. ) )
109	63, 107, 108	sylanbrc 698	. . . . 5 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( ( N ` { Z } ) \ { .0. } ) )
110		simp3 1063	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) )
111	2, 3	lmodvacl 18877	. . . . . . . . 9 |- ( ( W e. LMod /\ ( k ( .s ` W ) Y ) e. V /\ ( l ( .s ` W ) Z ) e. V ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. V )
112	15, 76, 41, 111	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. V )
113	2, 7	lspsnid 18993	. . . . . . . 8 |- ( ( W e. LMod /\ ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. V ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) )
114	15, 112, 113	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) )
115	110, 114	eqeltrd 2701	. . . . . 6 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> X e. ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) )
116	2, 4, 6, 5, 32, 7	lspsnvs 19114	. . . . . . . 8 |- ( ( W e. LVec /\ ( ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) /\ ( ( invr ` (Scalar ` W ) ) ` k ) =/= ( 0g ` (Scalar ` W ) ) ) /\ ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) e. V ) -> ( N ` { ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) } ) = ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) )
117	20, 59, 65, 112, 116	syl121anc 1331	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) } ) = ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) )
118	2, 3, 4, 6, 5	lmodvsdi 18886	. . . . . . . . . . 11 |- ( ( W e. LMod /\ ( ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) /\ ( k ( .s ` W ) Y ) e. V /\ ( l ( .s ` W ) Z ) e. V ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) = ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) )
119	15, 59, 76, 41, 118	syl13anc 1328	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) = ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) )
120		eqid 2622	. . . . . . . . . . . . . . 15 |- ( .r ` (Scalar ` W ) ) = ( .r ` (Scalar ` W ) )
121		eqid 2622	. . . . . . . . . . . . . . 15 |- ( 1r ` (Scalar ` W ) ) = ( 1r ` (Scalar ` W ) )
122	5, 32, 120, 121, 57	drnginvrl 18766	. . . . . . . . . . . . . 14 |- ( ( (Scalar ` W ) e. DivRing /\ k e. ( Base ` (Scalar ` W ) ) /\ k =/= ( 0g ` (Scalar ` W ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .r ` (Scalar ` W ) ) k ) = ( 1r ` (Scalar ` W ) ) )
123	22, 23, 56, 122	syl3anc 1326	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .r ` (Scalar ` W ) ) k ) = ( 1r ` (Scalar ` W ) ) )
124	123	oveq1d 6665	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .r ` (Scalar ` W ) ) k ) ( .s ` W ) Y ) = ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) )
125	2, 4, 6, 5, 120	lmodvsass 18888	. . . . . . . . . . . . 13 |- ( ( W e. LMod /\ ( ( ( invr ` (Scalar ` W ) ) ` k ) e. ( Base ` (Scalar ` W ) ) /\ k e. ( Base ` (Scalar ` W ) ) /\ Y e. V ) ) -> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .r ` (Scalar ` W ) ) k ) ( .s ` W ) Y ) = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) )
126	15, 59, 23, 74, 125	syl13anc 1328	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .r ` (Scalar ` W ) ) k ) ( .s ` W ) Y ) = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) )
127	2, 4, 6, 121	lmodvs1 18891	. . . . . . . . . . . . 13 |- ( ( W e. LMod /\ Y e. V ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) = Y )
128	15, 74, 127	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( 1r ` (Scalar ` W ) ) ( .s ` W ) Y ) = Y )
129	124, 126, 128	3eqtr3d 2664	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) = Y )
130	129	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( k ( .s ` W ) Y ) ) .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) = ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) )
131	119, 130	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) = ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) )
132	131	sneqd 4189	. . . . . . . 8 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> { ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) } = { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } )
133	132	fveq2d 6195	. . . . . . 7 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) } ) = ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) )
134	117, 133	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> ( N ` { ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) } ) = ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) )
135	115, 134	eleqtrd 2703	. . . . 5 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> X e. ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) )
136		oveq2 6658	. . . . . . . . 9 |- ( z = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) -> ( Y .+ z ) = ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) )
137	136	sneqd 4189	. . . . . . . 8 |- ( z = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) -> { ( Y .+ z ) } = { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } )
138	137	fveq2d 6195	. . . . . . 7 |- ( z = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) -> ( N ` { ( Y .+ z ) } ) = ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) )
139	138	eleq2d 2687	. . . . . 6 |- ( z = ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) -> ( X e. ( N ` { ( Y .+ z ) } ) <-> X e. ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) ) )
140	139	rspcev 3309	. . . . 5 |- ( ( ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) e. ( ( N ` { Z } ) \ { .0. } ) /\ X e. ( N ` { ( Y .+ ( ( ( invr ` (Scalar ` W ) ) ` k ) ( .s ` W ) ( l ( .s ` W ) Z ) ) ) } ) ) -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) )
141	109, 135, 140	syl2anc 693	. . . 4 |- ( ( ph /\ ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) /\ X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) ) -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) )
142	141	3exp 1264	. . 3 |- ( ph -> ( ( k e. ( Base ` (Scalar ` W ) ) /\ l e. ( Base ` (Scalar ` W ) ) ) -> ( X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) ) ) )
143	142	rexlimdvv 3037	. 2 |- ( ph -> ( E. k e. ( Base ` (Scalar ` W ) ) E. l e. ( Base ` (Scalar ` W ) ) X = ( ( k ( .s ` W ) Y ) .+ ( l ( .s ` W ) Z ) ) -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) ) )
144	14, 143	mpd 15	1 |- ( ph -> E. z e. ( ( N ` { Z } ) \ { .0. } ) X e. ( N ` { ( Y .+ z ) } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ 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   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   1rcur 18501   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:  lsatfixedN  34296