Theorem ipeq0 19983

Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
phlsrng.f	|- F = (Scalar ` W )
phllmhm.h	|- ., = ( .i ` W )
phllmhm.v	|- V = ( Base ` W )
ip0l.z	|- Z = ( 0g ` F )
ip0l.o	|- .0. = ( 0g ` W )

Assertion

Ref	Expression
ipeq0	|- ( ( W e. PreHil /\ A e. V ) -> ( ( A ., A ) = Z <-> A = .0. ) )

Proof of Theorem ipeq0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phllmhm.v	. . . . . 6 |- V = ( Base ` W )
2		phlsrng.f	. . . . . 6 |- F = (Scalar ` W )
3		phllmhm.h	. . . . . 6 |- ., = ( .i ` W )
4		ip0l.o	. . . . . 6 |- .0. = ( 0g ` W )
5		eqid 2622	. . . . . 6 |- ( *r ` F ) = ( *r ` F )
6		ip0l.z	. . . . . 6 |- Z = ( 0g ` F )
7	1, 2, 3, 4, 5, 6	isphl 19973	. . . . 5 |- ( W e. PreHil <-> ( W e. LVec /\ F e. *Ring /\ A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) ) )
8	7	simp3bi 1078	. . . 4 |- ( W e. PreHil -> A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) )
9		simp2 1062	. . . . 5 |- ( ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) -> ( ( x ., x ) = Z -> x = .0. ) )
10	9	ralimi 2952	. . . 4 |- ( A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom (ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) -> A. x e. V ( ( x ., x ) = Z -> x = .0. ) )
11	8, 10	syl 17	. . 3 |- ( W e. PreHil -> A. x e. V ( ( x ., x ) = Z -> x = .0. ) )
12		oveq12 6659	. . . . . . 7 |- ( ( x = A /\ x = A ) -> ( x ., x ) = ( A ., A ) )
13	12	anidms 677	. . . . . 6 |- ( x = A -> ( x ., x ) = ( A ., A ) )
14	13	eqeq1d 2624	. . . . 5 |- ( x = A -> ( ( x ., x ) = Z <-> ( A ., A ) = Z ) )
15		eqeq1 2626	. . . . 5 |- ( x = A -> ( x = .0. <-> A = .0. ) )
16	14, 15	imbi12d 334	. . . 4 |- ( x = A -> ( ( ( x ., x ) = Z -> x = .0. ) <-> ( ( A ., A ) = Z -> A = .0. ) ) )
17	16	rspccva 3308	. . 3 |- ( ( A. x e. V ( ( x ., x ) = Z -> x = .0. ) /\ A e. V ) -> ( ( A ., A ) = Z -> A = .0. ) )
18	11, 17	sylan 488	. 2 |- ( ( W e. PreHil /\ A e. V ) -> ( ( A ., A ) = Z -> A = .0. ) )
19	2, 3, 1, 6, 4	ip0l 19981	. . 3 |- ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = Z )
20		oveq1 6657	. . . 4 |- ( A = .0. -> ( A ., A ) = ( .0. ., A ) )
21	20	eqeq1d 2624	. . 3 |- ( A = .0. -> ( ( A ., A ) = Z <-> ( .0. ., A ) = Z ) )
22	19, 21	syl5ibrcom 237	. 2 |- ( ( W e. PreHil /\ A e. V ) -> ( A = .0. -> ( A ., A ) = Z ) )
23	18, 22	impbid 202	1 |- ( ( W e. PreHil /\ A e. V ) -> ( ( A ., A ) = Z <-> A = .0. ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   Basecbs 15857   *rcstv 15943  Scalarcsca 15944   .icip 15946   0gc0g 16100   *Ringcsr 18844   LMHom clmhm 19019   LVecclvec 19102  ringLModcrglmod 19169   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-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-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-sca 15957  df-vsca 15958  df-ip 15959  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-ghm 17658  df-lmod 18865  df-lmhm 19022  df-lvec 19103  df-sra 19172  df-rgmod 19173  df-phl 19971

This theorem is referenced by:  ip2eq  19998  ocvin  20018  lsmcss  20036  obsne0  20069  cphipeq0  23004  ipcau2  23033  tchcph  23036