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Theorem axhfvadd-zf 26097
Description: Derive axiom ax-hfvadd 26115 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
axhfvadd-zf  |-  +h  :
( ~H  X.  ~H )
--> ~H

Proof of Theorem axhfvadd-zf
StepHypRef Expression
1 axhil.2 . 2  |-  U  e. 
CHilOLD
2 df-hba 26084 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5851 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2486 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 26006 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 26089 . . 3  |-  +h  =  ( +v `  U )
85, 7hladdf 26013 . 2  |-  ( U  e.  CHilOLD  ->  +h  : ( ~H  X.  ~H ) --> ~H )
91, 8ax-mp 5 1  |-  +h  :
( ~H  X.  ~H )
--> ~H
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823   <.cop 4022    X. cxp 4986   -->wf 5566   ` cfv 5570   BaseSetcba 25677   CHilOLDchlo 25999   ~Hchil 26034    +h cva 26035    .h csm 26036   normhcno 26038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-1st 6773  df-2nd 6774  df-grpo 25391  df-ablo 25482  df-vc 25637  df-nv 25683  df-va 25686  df-ba 25687  df-sm 25688  df-0v 25689  df-nmcv 25691  df-cbn 25977  df-hlo 26000  df-hba 26084
This theorem is referenced by: (None)
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