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Theorem axhvass-zf 25605
Description: Derive axiom ax-hvass 25623 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 25590 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5869 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2499 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 25512 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 25595 . . 3  |-  +h  =  ( +v `  U )
85, 7hlass 25521 . 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 670 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 973    = wceq 1379    e. wcel 1767   <.cop 4033   ` cfv 5588  (class class class)co 6284   BaseSetcba 25183   CHilOLDchlo 25505   ~Hchil 25540    +h cva 25541    .h csm 25542   normhcno 25544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-1st 6784  df-2nd 6785  df-grpo 24897  df-ablo 24988  df-vc 25143  df-nv 25189  df-va 25192  df-ba 25193  df-sm 25194  df-0v 25195  df-nmcv 25197  df-cbn 25483  df-hlo 25506  df-hba 25590
This theorem is referenced by: (None)
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