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Theorem hial2eq 21701
Description: Two vectors whose inner product is always equal are equal. (Contributed by NM, 16-Nov-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hial2eq  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( A  .ih  x )  =  ( B  .ih  x )  <-> 
A  =  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem hial2eq
StepHypRef Expression
1 hvsubcl 21613 . . . 4  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A  -h  B
)  e.  ~H )
2 oveq2 5882 . . . . . 6  |-  ( x  =  ( A  -h  B )  ->  ( A  .ih  x )  =  ( A  .ih  ( A  -h  B ) ) )
3 oveq2 5882 . . . . . 6  |-  ( x  =  ( A  -h  B )  ->  ( B  .ih  x )  =  ( B  .ih  ( A  -h  B ) ) )
42, 3eqeq12d 2310 . . . . 5  |-  ( x  =  ( A  -h  B )  ->  (
( A  .ih  x
)  =  ( B 
.ih  x )  <->  ( A  .ih  ( A  -h  B
) )  =  ( B  .ih  ( A  -h  B ) ) ) )
54rspcv 2893 . . . 4  |-  ( ( A  -h  B )  e.  ~H  ->  ( A. x  e.  ~H  ( A  .ih  x )  =  ( B  .ih  x )  ->  ( A  .ih  ( A  -h  B ) )  =  ( B  .ih  ( A  -h  B ) ) ) )
61, 5syl 15 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( A  .ih  x )  =  ( B  .ih  x )  ->  ( A  .ih  ( A  -h  B
) )  =  ( B  .ih  ( A  -h  B ) ) ) )
7 hi2eq 21700 . . 3  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( ( A  .ih  ( A  -h  B
) )  =  ( B  .ih  ( A  -h  B ) )  <-> 
A  =  B ) )
86, 7sylibd 205 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( A  .ih  x )  =  ( B  .ih  x )  ->  A  =  B ) )
9 oveq1 5881 . . 3  |-  ( A  =  B  ->  ( A  .ih  x )  =  ( B  .ih  x
) )
109ralrimivw 2640 . 2  |-  ( A  =  B  ->  A. x  e.  ~H  ( A  .ih  x )  =  ( B  .ih  x ) )
118, 10impbid1 194 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H )  ->  ( A. x  e. 
~H  ( A  .ih  x )  =  ( B  .ih  x )  <-> 
A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556  (class class class)co 5874   ~Hchil 21515    .ih csp 21518    -h cmv 21521
This theorem is referenced by:  hial2eq2  21702  hoeq1  22426  hoeq2  22427  unoplin  22516  hmoplin  22538  pjss2coi  22760  pj3cor1i  22805
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-hfvadd 21596  ax-hvcom 21597  ax-hvass 21598  ax-hv0cl 21599  ax-hvaddid 21600  ax-hfvmul 21601  ax-hvmulid 21602  ax-hvdistr2 21605  ax-hvmul0 21606  ax-hfi 21674  ax-his2 21678  ax-his3 21679  ax-his4 21680
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-hvsub 21567
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