Theorem lshpkrlem1 34397

Description: Lemma for lshpkrex 34405. The value of tentative functional G is zero iff its argument belongs to hyperplane U. (Contributed by NM, 14-Jul-2014.)

Hypotheses

Ref	Expression
lshpkrlem.v	|- V = ( Base ` W )
lshpkrlem.a	|- .+ = ( +g ` W )
lshpkrlem.n	|- N = ( LSpan ` W )
lshpkrlem.p	|- .(+) = ( LSSum ` W )
lshpkrlem.h	|- H = (LSHyp ` W )
lshpkrlem.w	|- ( ph -> W e. LVec )
lshpkrlem.u	|- ( ph -> U e. H )
lshpkrlem.z	|- ( ph -> Z e. V )
lshpkrlem.x	|- ( ph -> X e. V )
lshpkrlem.e	|- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
lshpkrlem.d	|- D = (Scalar ` W )
lshpkrlem.k	|- K = ( Base ` D )
lshpkrlem.t	|- .x. = ( .s ` W )
lshpkrlem.o	|- .0. = ( 0g ` D )
lshpkrlem.g	|- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )

Assertion

Ref	Expression
lshpkrlem1	|- ( ph -> ( X e. U <-> ( G ` X ) = .0. ) )

Distinct variable groups:   x, k, y, .+   k, K, x   .0. , k   .x. , k, x, y   U, k, x, y   x, V   k, X, x, y   k, Z, x, y

Allowed substitution hints:   ph( x, y, k)   D( x, y, k)   .(+) ( x, y, k)   G( x, y, k)   H( x, y, k)   K( y)   N( x, y, k)   V( y, k)   W( x, y, k)   .0. ( x, y)

Proof of Theorem lshpkrlem1

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lshpkrlem.w	. . . . 5 |- ( ph -> W e. LVec )
2		lveclmod 19106	. . . . 5 |- ( W e. LVec -> W e. LMod )
3	1, 2	syl 17	. . . 4 |- ( ph -> W e. LMod )
4		lshpkrlem.d	. . . . 5 |- D = (Scalar ` W )
5	4	lmodfgrp 18872	. . . 4 |- ( W e. LMod -> D e. Grp )
6		lshpkrlem.k	. . . . 5 |- K = ( Base ` D )
7		lshpkrlem.o	. . . . 5 |- .0. = ( 0g ` D )
8	6, 7	grpidcl 17450	. . . 4 |- ( D e. Grp -> .0. e. K )
9	3, 5, 8	3syl 18	. . 3 |- ( ph -> .0. e. K )
10		lshpkrlem.v	. . . 4 |- V = ( Base ` W )
11		lshpkrlem.a	. . . 4 |- .+ = ( +g ` W )
12		lshpkrlem.n	. . . 4 |- N = ( LSpan ` W )
13		lshpkrlem.p	. . . 4 |- .(+) = ( LSSum ` W )
14		lshpkrlem.h	. . . 4 |- H = (LSHyp ` W )
15		lshpkrlem.u	. . . 4 |- ( ph -> U e. H )
16		lshpkrlem.z	. . . 4 |- ( ph -> Z e. V )
17		lshpkrlem.x	. . . 4 |- ( ph -> X e. V )
18		lshpkrlem.e	. . . 4 |- ( ph -> ( U .(+) ( N ` { Z } ) ) = V )
19		lshpkrlem.t	. . . 4 |- .x. = ( .s ` W )
20	10, 11, 12, 13, 14, 1, 15, 16, 17, 18, 4, 6, 19	lshpsmreu 34396	. . 3 |- ( ph -> E! k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) )
21		oveq1 6657	. . . . . . 7 |- ( k = .0. -> ( k .x. Z ) = ( .0. .x. Z ) )
22	21	oveq2d 6666	. . . . . 6 |- ( k = .0. -> ( b .+ ( k .x. Z ) ) = ( b .+ ( .0. .x. Z ) ) )
23	22	eqeq2d 2632	. . . . 5 |- ( k = .0. -> ( X = ( b .+ ( k .x. Z ) ) <-> X = ( b .+ ( .0. .x. Z ) ) ) )
24	23	rexbidv 3052	. . . 4 |- ( k = .0. -> ( E. b e. U X = ( b .+ ( k .x. Z ) ) <-> E. b e. U X = ( b .+ ( .0. .x. Z ) ) ) )
25	24	riota2 6633	. . 3 |- ( ( .0. e. K /\ E! k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) -> ( E. b e. U X = ( b .+ ( .0. .x. Z ) ) <-> ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) = .0. ) )
26	9, 20, 25	syl2anc 693	. 2 |- ( ph -> ( E. b e. U X = ( b .+ ( .0. .x. Z ) ) <-> ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) = .0. ) )
27		simpr 477	. . . . . 6 |- ( ( ph /\ X e. U ) -> X e. U )
28		eqidd 2623	. . . . . 6 |- ( ( ph /\ X e. U ) -> X = X )
29		eqeq2 2633	. . . . . . 7 |- ( b = X -> ( X = b <-> X = X ) )
30	29	rspcev 3309	. . . . . 6 |- ( ( X e. U /\ X = X ) -> E. b e. U X = b )
31	27, 28, 30	syl2anc 693	. . . . 5 |- ( ( ph /\ X e. U ) -> E. b e. U X = b )
32	31	ex 450	. . . 4 |- ( ph -> ( X e. U -> E. b e. U X = b ) )
33		eleq1a 2696	. . . . . 6 |- ( b e. U -> ( X = b -> X e. U ) )
34	33	a1i 11	. . . . 5 |- ( ph -> ( b e. U -> ( X = b -> X e. U ) ) )
35	34	rexlimdv 3030	. . . 4 |- ( ph -> ( E. b e. U X = b -> X e. U ) )
36	32, 35	impbid 202	. . 3 |- ( ph -> ( X e. U <-> E. b e. U X = b ) )
37		eqid 2622	. . . . . . . . . . 11 |- ( 0g ` W ) = ( 0g ` W )
38	10, 4, 19, 7, 37	lmod0vs 18896	. . . . . . . . . 10 |- ( ( W e. LMod /\ Z e. V ) -> ( .0. .x. Z ) = ( 0g ` W ) )
39	3, 16, 38	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( .0. .x. Z ) = ( 0g ` W ) )
40	39	adantr 481	. . . . . . . 8 |- ( ( ph /\ b e. U ) -> ( .0. .x. Z ) = ( 0g ` W ) )
41	40	oveq2d 6666	. . . . . . 7 |- ( ( ph /\ b e. U ) -> ( b .+ ( .0. .x. Z ) ) = ( b .+ ( 0g ` W ) ) )
42	1	adantr 481	. . . . . . . . 9 |- ( ( ph /\ b e. U ) -> W e. LVec )
43	42, 2	syl 17	. . . . . . . 8 |- ( ( ph /\ b e. U ) -> W e. LMod )
44		eqid 2622	. . . . . . . . . 10 |- ( LSubSp ` W ) = ( LSubSp ` W )
45	44, 14, 3, 15	lshplss 34268	. . . . . . . . 9 |- ( ph -> U e. ( LSubSp ` W ) )
46	10, 44	lssel 18938	. . . . . . . . 9 |- ( ( U e. ( LSubSp ` W ) /\ b e. U ) -> b e. V )
47	45, 46	sylan 488	. . . . . . . 8 |- ( ( ph /\ b e. U ) -> b e. V )
48	10, 11, 37	lmod0vrid 18894	. . . . . . . 8 |- ( ( W e. LMod /\ b e. V ) -> ( b .+ ( 0g ` W ) ) = b )
49	43, 47, 48	syl2anc 693	. . . . . . 7 |- ( ( ph /\ b e. U ) -> ( b .+ ( 0g ` W ) ) = b )
50	41, 49	eqtrd 2656	. . . . . 6 |- ( ( ph /\ b e. U ) -> ( b .+ ( .0. .x. Z ) ) = b )
51	50	eqeq2d 2632	. . . . 5 |- ( ( ph /\ b e. U ) -> ( X = ( b .+ ( .0. .x. Z ) ) <-> X = b ) )
52	51	bicomd 213	. . . 4 |- ( ( ph /\ b e. U ) -> ( X = b <-> X = ( b .+ ( .0. .x. Z ) ) ) )
53	52	rexbidva 3049	. . 3 |- ( ph -> ( E. b e. U X = b <-> E. b e. U X = ( b .+ ( .0. .x. Z ) ) ) )
54	36, 53	bitrd 268	. 2 |- ( ph -> ( X e. U <-> E. b e. U X = ( b .+ ( .0. .x. Z ) ) ) )
55		eqeq1 2626	. . . . . . . 8 |- ( x = X -> ( x = ( y .+ ( k .x. Z ) ) <-> X = ( y .+ ( k .x. Z ) ) ) )
56	55	rexbidv 3052	. . . . . . 7 |- ( x = X -> ( E. y e. U x = ( y .+ ( k .x. Z ) ) <-> E. y e. U X = ( y .+ ( k .x. Z ) ) ) )
57	56	riotabidv 6613	. . . . . 6 |- ( x = X -> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) = ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) )
58		lshpkrlem.g	. . . . . 6 |- G = ( x e. V |-> ( iota_ k e. K E. y e. U x = ( y .+ ( k .x. Z ) ) ) )
59		riotaex 6615	. . . . . 6 |- ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) e. _V
60	57, 58, 59	fvmpt 6282	. . . . 5 |- ( X e. V -> ( G ` X ) = ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) )
61		oveq1 6657	. . . . . . . . 9 |- ( y = b -> ( y .+ ( k .x. Z ) ) = ( b .+ ( k .x. Z ) ) )
62	61	eqeq2d 2632	. . . . . . . 8 |- ( y = b -> ( X = ( y .+ ( k .x. Z ) ) <-> X = ( b .+ ( k .x. Z ) ) ) )
63	62	cbvrexv 3172	. . . . . . 7 |- ( E. y e. U X = ( y .+ ( k .x. Z ) ) <-> E. b e. U X = ( b .+ ( k .x. Z ) ) )
64	63	a1i 11	. . . . . 6 |- ( k e. K -> ( E. y e. U X = ( y .+ ( k .x. Z ) ) <-> E. b e. U X = ( b .+ ( k .x. Z ) ) ) )
65	64	riotabiia 6628	. . . . 5 |- ( iota_ k e. K E. y e. U X = ( y .+ ( k .x. Z ) ) ) = ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) )
66	60, 65	syl6eq 2672	. . . 4 |- ( X e. V -> ( G ` X ) = ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) )
67	17, 66	syl 17	. . 3 |- ( ph -> ( G ` X ) = ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) )
68	67	eqeq1d 2624	. 2 |- ( ph -> ( ( G ` X ) = .0. <-> ( iota_ k e. K E. b e. U X = ( b .+ ( k .x. Z ) ) ) = .0. ) )
69	26, 54, 68	3bitr4d 300	1 |- ( ph -> ( X e. U <-> ( G ` X ) = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   E!wreu 2914   {csn 4177   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   Grpcgrp 17422   LSSumclsm 18049   LModclmod 18863   LSubSpclss 18932   LSpanclspn 18971   LVecclvec 19102  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-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  df-lshyp 34264

This theorem is referenced by:  lshpkr  34404