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Theorem axhvass-zf 26027
Description: Derive axiom ax-hvass 26045 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 26012 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5875 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2489 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 25934 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 26017 . . 3  |-  +h  =  ( +v `  U )
85, 7hlass 25943 . 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 1395    e. wcel 1819   <.cop 4038   ` cfv 5594  (class class class)co 6296   BaseSetcba 25605   CHilOLDchlo 25927   ~Hchil 25962    +h cva 25963    .h csm 25964   normhcno 25966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-1st 6799  df-2nd 6800  df-grpo 25319  df-ablo 25410  df-vc 25565  df-nv 25611  df-va 25614  df-ba 25615  df-sm 25616  df-0v 25617  df-nmcv 25619  df-cbn 25905  df-hlo 25928  df-hba 26012
This theorem is referenced by: (None)
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