Theorem islshpat 34304

Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 34267. (Contributed by NM, 11-Jan-2015.)

Hypotheses

Ref	Expression
islshpat.v	|- V = ( Base ` W )
islshpat.s	|- S = ( LSubSp ` W )
islshpat.p	|- .(+) = ( LSSum ` W )
islshpat.h	|- H = (LSHyp ` W )
islshpat.a	|- A = (LSAtoms ` W )
islshpat.w	|- ( ph -> W e. LMod )

Assertion

Ref	Expression
islshpat	|- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) ) )

Distinct variable groups:   .(+) , q   S, q   U, q   V, q   W, q   ph, q

Allowed substitution hints:   A( q)   H( q)

Proof of Theorem islshpat

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		islshpat.v	. . 3 |- V = ( Base ` W )
2		eqid 2622	. . 3 |- ( LSpan ` W ) = ( LSpan ` W )
3		islshpat.s	. . 3 |- S = ( LSubSp ` W )
4		islshpat.p	. . 3 |- .(+) = ( LSSum ` W )
5		islshpat.h	. . 3 |- H = (LSHyp ` W )
6		islshpat.w	. . 3 |- ( ph -> W e. LMod )
7	1, 2, 3, 4, 5, 6	islshpsm 34267	. 2 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
8		df-3an 1039	. . . . 5 |- ( ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
9		r19.42v 3092	. . . . 5 |- ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
10	8, 9	bitr4i 267	. . . 4 |- ( ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
11		df-rex 2918	. . . . . . . 8 |- ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. v ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
12		simpr 477	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> v = ( 0g ` W ) )
13	12	sneqd 4189	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> { v } = { ( 0g ` W ) } )
14	13	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = ( ( LSpan ` W ) ` { ( 0g ` W ) } ) )
15	6	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> W e. LMod )
16		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0g ` W ) = ( 0g ` W )
17	16, 2	lspsn0 19008	. . . . . . . . . . . . . . . . . . . 20 |- ( W e. LMod -> ( ( LSpan ` W ) ` { ( 0g ` W ) } ) = { ( 0g ` W ) } )
18	15, 17	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { ( 0g ` W ) } ) = { ( 0g ` W ) } )
19	14, 18	eqtrd 2656	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = { ( 0g ` W ) } )
20	19	oveq2d 6666	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = ( U .(+) { ( 0g ` W ) } ) )
21		simplrl 800	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. S )
22	3	lsssubg 18957	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
23	15, 21, 22	syl2anc 693	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. (SubGrp ` W ) )
24	16, 4	lsm01 18084	. . . . . . . . . . . . . . . . . 18 |- ( U e. (SubGrp ` W ) -> ( U .(+) { ( 0g ` W ) } ) = U )
25	23, 24	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) { ( 0g ` W ) } ) = U )
26	20, 25	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = U )
27		simplrr 801	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U =/= V )
28	26, 27	eqnetrd 2861	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V )
29	28	ex 450	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( v = ( 0g ` W ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V ) )
30	29	necon2d 2817	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V -> v =/= ( 0g ` W ) ) )
31	30	pm4.71rd 667	. . . . . . . . . . . 12 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V <-> ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
32	31	pm5.32da 673	. . . . . . . . . . 11 |- ( ( ph /\ v e. V ) -> ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
33	32	pm5.32da 673	. . . . . . . . . 10 |- ( ph -> ( ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) ) )
34		eldifsn 4317	. . . . . . . . . . . 12 |- ( v e. ( V \ { ( 0g ` W ) } ) <-> ( v e. V /\ v =/= ( 0g ` W ) ) )
35	34	anbi1i 731	. . . . . . . . . . 11 |- ( ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( ( v e. V /\ v =/= ( 0g ` W ) ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
36		anass 681	. . . . . . . . . . . 12 |- ( ( ( v e. V /\ v =/= ( 0g ` W ) ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( v e. V /\ ( v =/= ( 0g ` W ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
37		an12 838	. . . . . . . . . . . . 13 |- ( ( v =/= ( 0g ` W ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
38	37	anbi2i 730	. . . . . . . . . . . 12 |- ( ( v e. V /\ ( v =/= ( 0g ` W ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) <-> ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
39	36, 38	bitri 264	. . . . . . . . . . 11 |- ( ( ( v e. V /\ v =/= ( 0g ` W ) ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
40	35, 39	bitr2i 265	. . . . . . . . . 10 |- ( ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( v =/= ( 0g ` W ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) <-> ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
41	33, 40	syl6bb 276	. . . . . . . . 9 |- ( ph -> ( ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
42	41	exbidv 1850	. . . . . . . 8 |- ( ph -> ( E. v ( v e. V /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> E. v ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
43	11, 42	syl5bb 272	. . . . . . 7 |- ( ph -> ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. v ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
44		fvex 6201	. . . . . . . . . 10 |- ( ( LSpan ` W ) ` { v } ) e. _V
45	44	rexcom4b 3227	. . . . . . . . 9 |- ( E. q E. v e. ( V \ { ( 0g ` W ) } ) ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) /\ q = ( ( LSpan ` W ) ` { v } ) ) <-> E. v e. ( V \ { ( 0g ` W ) } ) ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
46		df-rex 2918	. . . . . . . . 9 |- ( E. v e. ( V \ { ( 0g ` W ) } ) ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. v ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
47	45, 46	bitr2i 265	. . . . . . . 8 |- ( E. v ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> E. q E. v e. ( V \ { ( 0g ` W ) } ) ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) /\ q = ( ( LSpan ` W ) ` { v } ) ) )
48		ancom 466	. . . . . . . . . 10 |- ( ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) /\ q = ( ( LSpan ` W ) ` { v } ) ) <-> ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
49	48	rexbii 3041	. . . . . . . . 9 |- ( E. v e. ( V \ { ( 0g ` W ) } ) ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) /\ q = ( ( LSpan ` W ) ` { v } ) ) <-> E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
50	49	exbii 1774	. . . . . . . 8 |- ( E. q E. v e. ( V \ { ( 0g ` W ) } ) ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) /\ q = ( ( LSpan ` W ) ` { v } ) ) <-> E. q E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
51	47, 50	bitri 264	. . . . . . 7 |- ( E. v ( v e. ( V \ { ( 0g ` W ) } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) <-> E. q E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
52	43, 51	syl6bb 276	. . . . . 6 |- ( ph -> ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. q E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) ) )
53		r19.41v 3089	. . . . . . . 8 |- ( E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) )
54		oveq2 6658	. . . . . . . . . . . 12 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( U .(+) q ) = ( U .(+) ( ( LSpan ` W ) ` { v } ) ) )
55	54	eqeq1d 2624	. . . . . . . . . . 11 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( ( U .(+) q ) = V <-> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
56	55	anbi2d 740	. . . . . . . . . 10 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
57	56	pm5.32i 669	. . . . . . . . 9 |- ( ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
58	57	rexbii 3041	. . . . . . . 8 |- ( E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
59	53, 58	bitr3i 266	. . . . . . 7 |- ( ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
60	59	exbii 1774	. . . . . 6 |- ( E. q ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> E. q E. v e. ( V \ { ( 0g ` W ) } ) ( q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
61	52, 60	syl6bbr 278	. . . . 5 |- ( ph -> ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. q ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) ) )
62		islshpat.a	. . . . . . . . 9 |- A = (LSAtoms ` W )
63	1, 2, 16, 62	islsat 34278	. . . . . . . 8 |- ( W e. LMod -> ( q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) ) )
64	6, 63	syl 17	. . . . . . 7 |- ( ph -> ( q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) ) )
65	64	anbi1d 741	. . . . . 6 |- ( ph -> ( ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) ) )
66	65	exbidv 1850	. . . . 5 |- ( ph -> ( E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> E. q ( E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) ) )
67	61, 66	bitr4d 271	. . . 4 |- ( ph -> ( E. v e. V ( ( U e. S /\ U =/= V ) /\ ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) ) )
68	10, 67	syl5bb 272	. . 3 |- ( ph -> ( ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) ) )
69		df-3an 1039	. . . 4 |- ( ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ E. q e. A ( U .(+) q ) = V ) )
70		r19.42v 3092	. . . . 5 |- ( E. q e. A ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) <-> ( ( U e. S /\ U =/= V ) /\ E. q e. A ( U .(+) q ) = V ) )
71		df-rex 2918	. . . . 5 |- ( E. q e. A ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) <-> E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) )
72	70, 71	bitr3i 266	. . . 4 |- ( ( ( U e. S /\ U =/= V ) /\ E. q e. A ( U .(+) q ) = V ) <-> E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) )
73	69, 72	bitr2i 265	. . 3 |- ( E. q ( q e. A /\ ( ( U e. S /\ U =/= V ) /\ ( U .(+) q ) = V ) ) <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) )
74	68, 73	syl6bb 276	. 2 |- ( ph -> ( ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) ) )
75	7, 74	bitrd 268	1 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   0gc0g 16100  SubGrpcsubg 17588   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971  LSAtomsclsa 34261  LSHypclsh 34262

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-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-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  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-lmod 18865  df-lss 18933  df-lsp 18972  df-lsatoms 34263  df-lshyp 34264

This theorem is referenced by:  islshpcv  34340