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Theorem axhvaddid-zf 22481
Description: Derive axiom ax-hvaddid 22499 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. 
CHil OLD
Assertion
Ref Expression
axhvaddid-zf  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )

Proof of Theorem axhvaddid-zf
StepHypRef Expression
1 axhil.2 . 2  |-  U  e. 
CHil OLD
2 df-hba 22464 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5723 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2458 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 22386 . . . 4  |-  U  e.  NrmCVec
73, 6h2hva 22469 . . 3  |-  +h  =  ( +v `  U )
8 df-h0v 22465 . . . 4  |-  0h  =  ( 0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
93fveq2i 5723 . . . 4  |-  ( 0vec `  U )  =  (
0vec `  <. <.  +h  ,  .h  >. ,  normh >. )
108, 9eqtr4i 2458 . . 3  |-  0h  =  ( 0vec `  U )
115, 7, 10hladdid 22397 . 2  |-  ( ( U  e.  CHil OLD  /\  A  e.  ~H )  ->  ( A  +h  0h )  =  A )
121, 11mpan 652 1  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   <.cop 3809   ` cfv 5446  (class class class)co 6073   BaseSetcba 22057   0veccn0v 22059   CHil
OLDchlo 22379   ~Hchil 22414    +h cva 22415    .h csm 22416   normhcno 22418   0hc0v 22419
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-1st 6341  df-2nd 6342  df-riota 6541  df-grpo 21771  df-gid 21772  df-ablo 21862  df-vc 22017  df-nv 22063  df-va 22066  df-ba 22067  df-sm 22068  df-0v 22069  df-nmcv 22071  df-cbn 22357  df-hlo 22380  df-hba 22464  df-h0v 22465
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