Theorem ip2eq 19998

Description: Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
ip2eq.h	|- ., = ( .i ` W )
ip2eq.v	|- V = ( Base ` W )

Assertion

Ref	Expression
ip2eq	|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A = B <-> A. x e. V ( x ., A ) = ( x ., B ) ) )

Distinct variable groups:   x, A   x, B   x, .,   x, V   x, W

Proof of Theorem ip2eq

Step	Hyp	Ref	Expression
1		oveq2 6658	. . 3 |- ( A = B -> ( x ., A ) = ( x ., B ) )
2	1	ralrimivw 2967	. 2 |- ( A = B -> A. x e. V ( x ., A ) = ( x ., B ) )
3		phllmod 19975	. . . . 5 |- ( W e. PreHil -> W e. LMod )
4		ip2eq.v	. . . . . 6 |- V = ( Base ` W )
5		eqid 2622	. . . . . 6 |- ( -g ` W ) = ( -g ` W )
6	4, 5	lmodvsubcl 18908	. . . . 5 |- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A ( -g ` W ) B ) e. V )
7	3, 6	syl3an1 1359	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A ( -g ` W ) B ) e. V )
8		oveq1 6657	. . . . . 6 |- ( x = ( A ( -g ` W ) B ) -> ( x ., A ) = ( ( A ( -g ` W ) B ) ., A ) )
9		oveq1 6657	. . . . . 6 |- ( x = ( A ( -g ` W ) B ) -> ( x ., B ) = ( ( A ( -g ` W ) B ) ., B ) )
10	8, 9	eqeq12d 2637	. . . . 5 |- ( x = ( A ( -g ` W ) B ) -> ( ( x ., A ) = ( x ., B ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) )
11	10	rspcv 3305	. . . 4 |- ( ( A ( -g ` W ) B ) e. V -> ( A. x e. V ( x ., A ) = ( x ., B ) -> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) )
12	7, 11	syl 17	. . 3 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A. x e. V ( x ., A ) = ( x ., B ) -> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) )
13		simp1 1061	. . . . . . 7 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> W e. PreHil )
14		simp2 1062	. . . . . . 7 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> A e. V )
15		simp3 1063	. . . . . . 7 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> B e. V )
16		eqid 2622	. . . . . . . 8 |- (Scalar ` W ) = (Scalar ` W )
17		ip2eq.h	. . . . . . . 8 |- ., = ( .i ` W )
18		eqid 2622	. . . . . . . 8 |- ( -g ` (Scalar ` W ) ) = ( -g ` (Scalar ` W ) )
19	16, 17, 4, 5, 18	ipsubdi 19988	. . . . . . 7 |- ( ( W e. PreHil /\ ( ( A ( -g ` W ) B ) e. V /\ A e. V /\ B e. V ) ) -> ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) )
20	13, 7, 14, 15, 19	syl13anc 1328	. . . . . 6 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) )
21	20	eqeq1d 2624	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` (Scalar ` W ) ) ) )
22		eqid 2622	. . . . . . 7 |- ( 0g ` (Scalar ` W ) ) = ( 0g ` (Scalar ` W ) )
23		eqid 2622	. . . . . . 7 |- ( 0g ` W ) = ( 0g ` W )
24	16, 17, 4, 22, 23	ipeq0 19983	. . . . . 6 |- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) )
25	13, 7, 24	syl2anc 693	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) )
26	21, 25	bitr3d 270	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) )
27	3	3ad2ant1 1082	. . . . . 6 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> W e. LMod )
28	16	lmodfgrp 18872	. . . . . 6 |- ( W e. LMod -> (Scalar ` W ) e. Grp )
29	27, 28	syl 17	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> (Scalar ` W ) e. Grp )
30		eqid 2622	. . . . . . 7 |- ( Base ` (Scalar ` W ) ) = ( Base ` (Scalar ` W ) )
31	16, 17, 4, 30	ipcl 19978	. . . . . 6 |- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V /\ A e. V ) -> ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` (Scalar ` W ) ) )
32	13, 7, 14, 31	syl3anc 1326	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` (Scalar ` W ) ) )
33	16, 17, 4, 30	ipcl 19978	. . . . . 6 |- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` (Scalar ` W ) ) )
34	13, 7, 15, 33	syl3anc 1326	. . . . 5 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` (Scalar ` W ) ) )
35	30, 22, 18	grpsubeq0 17501	. . . . 5 |- ( ( (Scalar ` W ) e. Grp /\ ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` (Scalar ` W ) ) /\ ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` (Scalar ` W ) ) ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) )
36	29, 32, 34, 35	syl3anc 1326	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` (Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` (Scalar ` W ) ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) )
37		lmodgrp 18870	. . . . . 6 |- ( W e. LMod -> W e. Grp )
38	3, 37	syl 17	. . . . 5 |- ( W e. PreHil -> W e. Grp )
39	4, 23, 5	grpsubeq0 17501	. . . . 5 |- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) = ( 0g ` W ) <-> A = B ) )
40	38, 39	syl3an1 1359	. . . 4 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) = ( 0g ` W ) <-> A = B ) )
41	26, 36, 40	3bitr3d 298	. . 3 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) <-> A = B ) )
42	12, 41	sylibd 229	. 2 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A. x e. V ( x ., A ) = ( x ., B ) -> A = B ) )
43	2, 42	impbid2 216	1 |- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A = B <-> A. x e. V ( x ., A ) = ( x ., B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Scalarcsca 15944   .icip 15946   0gc0g 16100   Grpcgrp 17422   -gcsg 17424   LModclmod 18863   PreHilcphl 19969

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-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-map 7859  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-ghm 17658  df-mgp 18490  df-ur 18502  df-ring 18549  df-oppr 18623  df-rnghom 18715  df-staf 18845  df-srng 18846  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971

This theorem is referenced by: (None)