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Theorem h2hva 30993
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 2737 . . . 4 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
21vafval 30622 . . 3 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5469 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2838 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 30630 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1142 . . . . 5 norm ∈ V
103, 9op1st 8022 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6909 . . 3 (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (1st ‘⟨ + , · ⟩)
128simp1i 1140 . . . 4 + ∈ V
138simp2i 1141 . . . 4 · ∈ V
1412, 13op1st 8022 . . 3 (1st ‘⟨ + , · ⟩) = +
152, 11, 143eqtrri 2770 . 2 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6909 . 2 ( +𝑣𝑈) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2768 1 + = ( +𝑣𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632  cfv 6561  1st c1st 8012  NrmCVeccnv 30603   +𝑣 cpv 30604   + cva 30939   · csm 30940  normcno 30942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-oprab 7435  df-1st 8014  df-vc 30578  df-nv 30611  df-va 30614
This theorem is referenced by:  h2hvs  30996  axhfvadd-zf  31001  axhvcom-zf  31002  axhvass-zf  31003  axhvaddid-zf  31005  axhvdistr1-zf  31009  axhvdistr2-zf  31010  axhis2-zf  31014  hhva  31185
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