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Theorem axhvdistr2-zf 26479
Description: Derive axiom ax-hvdistr2 26497 from Hilbert space under ZF set theory. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
axhil.1  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
axhil.2  |-  U  e. 
CHilOLD
Assertion
Ref Expression
axhvdistr2-zf  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )

Proof of Theorem axhvdistr2-zf
StepHypRef Expression
1 axhil.2 . 2  |-  U  e. 
CHilOLD
2 df-hba 26457 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5884 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2461 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 26379 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 26462 . . 3  |-  +h  =  ( +v `  U )
83, 6h2hsm 26463 . . 3  |-  .h  =  ( .sOLD `  U
)
95, 7, 8hldir 26395 . 2  |-  ( ( U  e.  CHilOLD  /\  ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )
)  ->  ( ( A  +  B )  .h  C )  =  ( ( A  .h  C
)  +h  ( B  .h  C ) ) )
101, 9mpan 674 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  ~H )  ->  (
( A  +  B
)  .h  C )  =  ( ( A  .h  C )  +h  ( B  .h  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1870   <.cop 4008   ` cfv 5601  (class class class)co 6305   CCcc 9536    + caddc 9541   BaseSetcba 26050   CHilOLDchlo 26372   ~Hchil 26407    +h cva 26408    .h csm 26409   normhcno 26411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-1st 6807  df-2nd 6808  df-vc 26010  df-nv 26056  df-va 26059  df-ba 26060  df-sm 26061  df-0v 26062  df-nmcv 26064  df-cbn 26350  df-hlo 26373  df-hba 26457
This theorem is referenced by: (None)
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