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Theorem axhvass-zf 26572
Description: Derive axiom ax-hvass 26590 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
axhvass-zf  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )

Proof of Theorem axhvass-zf
StepHypRef Expression
1 axhil.2 . 2  |-  U  e. 
CHilOLD
2 df-hba 26557 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5821 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2447 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 26479 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 26562 . . 3  |-  +h  =  ( +v `  U )
85, 7hlass 26488 . 2  |-  ( ( U  e.  CHilOLD  /\  ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )
)  ->  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) ) )
91, 8mpan 674 1  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1872   <.cop 3940   ` cfv 5537  (class class class)co 6242   BaseSetcba 26140   CHilOLDchlo 26472   ~Hchil 26507    +h cva 26508    .h csm 26509   normhcno 26511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-ral 2713  df-rex 2714  df-reu 2715  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3698  df-if 3848  df-sn 3935  df-pr 3937  df-op 3941  df-uni 4156  df-iun 4237  df-br 4360  df-opab 4419  df-mpt 4420  df-id 4704  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-ov 6245  df-oprab 6246  df-1st 6744  df-2nd 6745  df-grpo 25854  df-ablo 25945  df-vc 26100  df-nv 26146  df-va 26149  df-ba 26150  df-sm 26151  df-0v 26152  df-nmcv 26154  df-cbn 26440  df-hlo 26473  df-hba 26557
This theorem is referenced by: (None)
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