HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  h2hva Structured version   Visualization version   GIF version

Theorem h2hva 27959
Description: The group (addition) operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
h2h.1 𝑈 = ⟨⟨ + , · ⟩, norm
h2h.2 𝑈 ∈ NrmCVec
Assertion
Ref Expression
h2hva + = ( +𝑣𝑈)

Proof of Theorem h2hva
StepHypRef Expression
1 eqid 2651 . . . 4 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
21vafval 27586 . . 3 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 4962 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2727 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 27594 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1092 . . . . 5 norm ∈ V
103, 9op1st 7218 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6232 . . 3 (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (1st ‘⟨ + , · ⟩)
128simp1i 1090 . . . 4 + ∈ V
138simp2i 1091 . . . 4 · ∈ V
1412, 13op1st 7218 . . 3 (1st ‘⟨ + , · ⟩) = +
152, 11, 143eqtrri 2678 . 2 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6232 . 2 ( +𝑣𝑈) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2676 1 + = ( +𝑣𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216  cfv 5926  1st c1st 7208  NrmCVeccnv 27567   +𝑣 cpv 27568   + cva 27905   · csm 27906  normcno 27908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-oprab 6694  df-1st 7210  df-vc 27542  df-nv 27575  df-va 27578
This theorem is referenced by:  h2hvs  27962  axhfvadd-zf  27967  axhvcom-zf  27968  axhvass-zf  27969  axhvaddid-zf  27971  axhvdistr1-zf  27975  axhvdistr2-zf  27976  axhis2-zf  27980  hhva  28151
  Copyright terms: Public domain W3C validator