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Theorem axhvmulid-zf 22448
Description: Derive axiom ax-hvmulid 22466 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
axhvmulid-zf  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )

Proof of Theorem axhvmulid-zf
StepHypRef Expression
1 axhil.2 . 2  |-  U  e. 
CHil OLD
2 df-hba 22429 . . . 4  |-  ~H  =  ( BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
3 axhil.1 . . . . 5  |-  U  = 
<. <.  +h  ,  .h  >. ,  normh >.
43fveq2i 5694 . . . 4  |-  ( BaseSet `  U )  =  (
BaseSet `  <. <.  +h  ,  .h  >. ,  normh >. )
52, 4eqtr4i 2431 . . 3  |-  ~H  =  ( BaseSet `  U )
61hlnvi 22351 . . . 4  |-  U  e.  NrmCVec
73, 6h2hsm 22435 . . 3  |-  .h  =  ( .s OLD `  U
)
85, 7hlmulid 22364 . 2  |-  ( ( U  e.  CHil OLD  /\  A  e.  ~H )  ->  ( 1  .h  A
)  =  A )
91, 8mpan 652 1  |-  ( A  e.  ~H  ->  (
1  .h  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   <.cop 3781   ` cfv 5417  (class class class)co 6044   1c1 8951   BaseSetcba 22022   CHil
OLDchlo 22344   ~Hchil 22379    +h cva 22380    .h csm 22381   normhcno 22383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-1st 6312  df-2nd 6313  df-vc 21982  df-nv 22028  df-va 22031  df-ba 22032  df-sm 22033  df-0v 22034  df-nmcv 22036  df-cbn 22322  df-hlo 22345  df-hba 22429
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