Theorem islshpat 29500

Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29463. (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 2404	. . 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 29463	. 2 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
8		df-3an 938	. . . . 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 2822	. . . . 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 244	. . . 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 2672	. . . . . . . 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 448	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> v = ( 0g ` W ) )
13	12	sneqd 3787	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> { v } = { ( 0g ` W ) } )
14	13	fveq2d 5691	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = ( ( LSpan ` W ) ` { ( 0g ` W ) } ) )
15	6	ad3antrrr 711	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> W e. LMod )
16		eqid 2404	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0g ` W ) = ( 0g ` W )
17	16, 2	lspsn0 16039	. . . . . . . . . . . . . . . . . . . 20 |- ( W e. LMod -> ( ( LSpan ` W ) ` { ( 0g ` W ) } ) = { ( 0g ` W ) } )
18	15, 17	syl 16	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { ( 0g ` W ) } ) = { ( 0g ` W ) } )
19	14, 18	eqtrd 2436	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = { ( 0g ` W ) } )
20	19	oveq2d 6056	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = ( U .(+) { ( 0g ` W ) } ) )
21		simplrl 737	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. S )
22	3	lsssubg 15988	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
23	15, 21, 22	syl2anc 643	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. (SubGrp ` W ) )
24	16, 4	lsm01 15258	. . . . . . . . . . . . . . . . . 18 |- ( U e. (SubGrp ` W ) -> ( U .(+) { ( 0g ` W ) } ) = U )
25	23, 24	syl 16	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) { ( 0g ` W ) } ) = U )
26	20, 25	eqtrd 2436	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = U )
27		simplrr 738	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U =/= V )
28	26, 27	eqnetrd 2585	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V )
29	28	ex 424	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( v = ( 0g ` W ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V ) )
30	29	necon2d 2617	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V -> v =/= ( 0g ` W ) ) )
31	30	pm4.71rd 617	. . . . . . . . . . . 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 623	. . . . . . . . . . 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 623	. . . . . . . . . 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 3887	. . . . . . . . . . . 12 |- ( v e. ( V \ { ( 0g ` W ) } ) <-> ( v e. V /\ v =/= ( 0g ` W ) ) )
35	34	anbi1i 677	. . . . . . . . . . 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 631	. . . . . . . . . . . 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 773	. . . . . . . . . . . . 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 676	. . . . . . . . . . . 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 241	. . . . . . . . . . 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 242	. . . . . . . . . 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 253	. . . . . . . . 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 1633	. . . . . . . 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 249	. . . . . . 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 5701	. . . . . . . . . 10 |- ( ( LSpan ` W ) ` { v } ) e. _V
45	44	rexcom4b 2937	. . . . . . . . 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 2672	. . . . . . . . 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 242	. . . . . . . 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 438	. . . . . . . . . 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 2691	. . . . . . . . 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 1589	. . . . . . . 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 241	. . . . . . 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 253	. . . . . 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 2821	. . . . . . . 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 6048	. . . . . . . . . . . 12 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( U .(+) q ) = ( U .(+) ( ( LSpan ` W ) ` { v } ) ) )
55	54	eqeq1d 2412	. . . . . . . . . . 11 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( ( U .(+) q ) = V <-> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
56	55	anbi2d 685	. . . . . . . . . 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 619	. . . . . . . . 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 2691	. . . . . . . 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 243	. . . . . . 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 1589	. . . . . 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 255	. . . . 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 29474	. . . . . . . 8 |- ( W e. LMod -> ( q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) ) )
64	6, 63	syl 16	. . . . . . 7 |- ( ph -> ( q e. A <-> E. v e. ( V \ { ( 0g ` W ) } ) q = ( ( LSpan ` W ) ` { v } ) ) )
65	64	anbi1d 686	. . . . . 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 1633	. . . . 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 248	. . . 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 249	. . 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 938	. . . 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 2822	. . . . 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 2672	. . . . 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 243	. . . 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 242	. . 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 253	. 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 245	1 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) ) )

Syntax hints:   -> wi 4   <-> wb 177   /\ wa 359   /\ w3a 936   E.wex 1547   = wceq 1649   e. wcel 1721   =/= wne 2567   E.wrex 2667   \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   0gc0g 13678  SubGrpcsubg 14893   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002  LSAtomsclsa 29457  LSHypclsh 29458

This theorem is referenced by:  islshpcv  29536

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lsatoms 29459  df-lshyp 29460