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Theorem h2hva 30696
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 2724 . . . 4 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
21vafval 30325 . . 3 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5454 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2822 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 30333 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1138 . . . . 5 norm ∈ V
103, 9op1st 7976 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6884 . . 3 (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (1st ‘⟨ + , · ⟩)
128simp1i 1136 . . . 4 + ∈ V
138simp2i 1137 . . . 4 · ∈ V
1412, 13op1st 7976 . . 3 (1st ‘⟨ + , · ⟩) = +
152, 11, 143eqtrri 2757 . 2 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6884 . 2 ( +𝑣𝑈) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2755 1 + = ( +𝑣𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  cop 4626  cfv 6533  1st c1st 7966  NrmCVeccnv 30306   +𝑣 cpv 30307   + cva 30642   · csm 30643  normcno 30645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fo 6539  df-fv 6541  df-oprab 7405  df-1st 7968  df-vc 30281  df-nv 30314  df-va 30317
This theorem is referenced by:  h2hvs  30699  axhfvadd-zf  30704  axhvcom-zf  30705  axhvass-zf  30706  axhvaddid-zf  30708  axhvdistr1-zf  30712  axhvdistr2-zf  30713  axhis2-zf  30717  hhva  30888
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