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Theorem hial2eq2 22566
Description: Two vectors whose inner product is always equal are equal. (Contributed by NM, 28-Jan-2006.) (New usage is discouraged.)
Assertion
Ref Expression
hial2eq2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( x  .ih  A )  =  ( x 
.ih  B )  <->  A  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem hial2eq2
StepHypRef Expression
1 ax-his1 22541 . . . . . 6  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( A  .ih  x
)  =  ( * `
 ( x  .ih  A ) ) )
2 ax-his1 22541 . . . . . 6  |-  ( ( B  e.  ~H  /\  x  e.  ~H )  ->  ( B  .ih  x
)  =  ( * `
 ( x  .ih  B ) ) )
31, 2eqeqan12d 2423 . . . . 5  |-  ( ( ( A  e.  ~H  /\  x  e.  ~H )  /\  ( B  e.  ~H  /\  x  e.  ~H )
)  ->  ( ( A  .ih  x )  =  ( B  .ih  x
)  <->  ( * `  ( x  .ih  A ) )  =  ( * `
 ( x  .ih  B ) ) ) )
4 hicl 22539 . . . . . . 7  |-  ( ( x  e.  ~H  /\  A  e.  ~H )  ->  ( x  .ih  A
)  e.  CC )
54ancoms 440 . . . . . 6  |-  ( ( A  e.  ~H  /\  x  e.  ~H )  ->  ( x  .ih  A
)  e.  CC )
6 hicl 22539 . . . . . . 7  |-  ( ( x  e.  ~H  /\  B  e.  ~H )  ->  ( x  .ih  B
)  e.  CC )
76ancoms 440 . . . . . 6  |-  ( ( B  e.  ~H  /\  x  e.  ~H )  ->  ( x  .ih  B
)  e.  CC )
8 cj11 11926 . . . . . 6  |-  ( ( ( x  .ih  A
)  e.  CC  /\  ( x  .ih  B )  e.  CC )  -> 
( ( * `  ( x  .ih  A ) )  =  ( * `
 ( x  .ih  B ) )  <->  ( x  .ih  A )  =  ( x  .ih  B ) ) )
95, 7, 8syl2an 464 . . . . 5  |-  ( ( ( A  e.  ~H  /\  x  e.  ~H )  /\  ( B  e.  ~H  /\  x  e.  ~H )
)  ->  ( (
* `  ( x  .ih  A ) )  =  ( * `  (
x  .ih  B )
)  <->  ( x  .ih  A )  =  ( x 
.ih  B ) ) )
103, 9bitr2d 246 . . . 4  |-  ( ( ( A  e.  ~H  /\  x  e.  ~H )  /\  ( B  e.  ~H  /\  x  e.  ~H )
)  ->  ( (
x  .ih  A )  =  ( x  .ih  B )  <->  ( A  .ih  x )  =  ( B  .ih  x ) ) )
1110anandirs 805 . . 3  |-  ( ( ( A  e.  ~H  /\  B  e.  ~H )  /\  x  e.  ~H )  ->  ( ( x 
.ih  A )  =  ( x  .ih  B
)  <->  ( A  .ih  x )  =  ( B  .ih  x ) ) )
1211ralbidva 2686 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( x  .ih  A )  =  ( x 
.ih  B )  <->  A. x  e.  ~H  ( A  .ih  x )  =  ( B  .ih  x ) ) )
13 hial2eq 22565 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( A  .ih  x )  =  ( B  .ih  x )  <-> 
A  =  B ) )
1412, 13bitrd 245 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( x  .ih  A )  =  ( x 
.ih  B )  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2670   ` cfv 5417  (class class class)co 6044   CCcc 8948   *ccj 11860   ~Hchil 22379    .ih csp 22382
This theorem is referenced by:  hoeq2  23291  adjvalval  23397  cnlnadjlem6  23532  adjlnop  23546  bra11  23568
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 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-hfvadd 22460  ax-hvcom 22461  ax-hvass 22462  ax-hv0cl 22463  ax-hvaddid 22464  ax-hfvmul 22465  ax-hvmulid 22466  ax-hvdistr2 22469  ax-hvmul0 22470  ax-hfi 22538  ax-his1 22541  ax-his2 22542  ax-his3 22543  ax-his4 22544
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-2 10018  df-cj 11863  df-re 11864  df-im 11865  df-hvsub 22431
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