Theorem islshpat 29877

Description: Hyperplane properties expressed with subspace sum and an atom. TODO: can proof be shortened? Seems long for a simple variation of islshpsm 29840. (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 2438	. . 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 29840	. 2 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. v e. V ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) ) )
8		df-3an 939	. . . . 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 2864	. . . . 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 245	. . . 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 2713	. . . . . . . 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 449	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> v = ( 0g ` W ) )
13	12	sneqd 3829	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> { v } = { ( 0g ` W ) } )
14	13	fveq2d 5734	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = ( ( LSpan ` W ) ` { ( 0g ` W ) } ) )
15	6	ad3antrrr 712	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> W e. LMod )
16		eqid 2438	. . . . . . . . . . . . . . . . . . . . 21 |- ( 0g ` W ) = ( 0g ` W )
17	16, 2	lspsn0 16086	. . . . . . . . . . . . . . . . . . . 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 2470	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( ( LSpan ` W ) ` { v } ) = { ( 0g ` W ) } )
20	19	oveq2d 6099	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = ( U .(+) { ( 0g ` W ) } ) )
21		simplrl 738	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. S )
22	3	lsssubg 16035	. . . . . . . . . . . . . . . . . . 19 |- ( ( W e. LMod /\ U e. S ) -> U e. (SubGrp ` W ) )
23	15, 21, 22	syl2anc 644	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U e. (SubGrp ` W ) )
24	16, 4	lsm01 15305	. . . . . . . . . . . . . . . . . 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 2470	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = U )
27		simplrr 739	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> U =/= V )
28	26, 27	eqnetrd 2621	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) /\ v = ( 0g ` W ) ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V )
29	28	ex 425	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( v = ( 0g ` W ) -> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) =/= V ) )
30	29	necon2d 2656	. . . . . . . . . . . . 13 |- ( ( ( ph /\ v e. V ) /\ ( U e. S /\ U =/= V ) ) -> ( ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V -> v =/= ( 0g ` W ) ) )
31	30	pm4.71rd 618	. . . . . . . . . . . 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 624	. . . . . . . . . . 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 624	. . . . . . . . . 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 3929	. . . . . . . . . . . 12 |- ( v e. ( V \ { ( 0g ` W ) } ) <-> ( v e. V /\ v =/= ( 0g ` W ) ) )
35	34	anbi1i 678	. . . . . . . . . . 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 632	. . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 677	. . . . . . . . . . . 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 242	. . . . . . . . . . 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 243	. . . . . . . . . 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 254	. . . . . . . . 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 1637	. . . . . . . 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 250	. . . . . . 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 5744	. . . . . . . . . 10 |- ( ( LSpan ` W ) ` { v } ) e. _V
45	44	rexcom4b 2979	. . . . . . . . 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 2713	. . . . . . . . 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 243	. . . . . . . 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 439	. . . . . . . . . 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 2732	. . . . . . . . 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 1593	. . . . . . . 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 242	. . . . . . 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 254	. . . . . 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 2863	. . . . . . . 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 6091	. . . . . . . . . . . 12 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( U .(+) q ) = ( U .(+) ( ( LSpan ` W ) ` { v } ) ) )
55	54	eqeq1d 2446	. . . . . . . . . . 11 |- ( q = ( ( LSpan ` W ) ` { v } ) -> ( ( U .(+) q ) = V <-> ( U .(+) ( ( LSpan ` W ) ` { v } ) ) = V ) )
56	55	anbi2d 686	. . . . . . . . . 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 620	. . . . . . . . 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 2732	. . . . . . . 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 244	. . . . . . 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 1593	. . . . . 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 256	. . . . 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 29851	. . . . . . . 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 687	. . . . . 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 1637	. . . . 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 249	. . . 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 250	. . 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 939	. . . 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 2864	. . . . 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 2713	. . . . 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 244	. . . 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 243	. . 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 254	. 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 246	1 |- ( ph -> ( U e. H <-> ( U e. S /\ U =/= V /\ E. q e. A ( U .(+) q ) = V ) ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ wa 360   /\ w3a 937   E.wex 1551   = wceq 1653   e. wcel 1726   =/= wne 2601   E.wrex 2708   \ cdif 3319   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471   0gc0g 13725  SubGrpcsubg 14940   LSSumclsm 15270   LModclmod 15952   LSubSpclss 16010   LSpanclspn 16049  LSAtomsclsa 29834  LSHypclsh 29835

This theorem is referenced by:  islshpcv  29913

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-0g 13729  df-mnd 14692  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-cntz 15118  df-lsm 15272  df-cmn 15416  df-abl 15417  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lsp 16050  df-lsatoms 29836  df-lshyp 29837