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Theorem h2hva 29965
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 2733 . . . 4 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
21vafval 29594 . . 3 ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩) = (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5425 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2831 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 29602 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1142 . . . . 5 norm ∈ V
103, 9op1st 7933 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6849 . . 3 (1st ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (1st ‘⟨ + , · ⟩)
128simp1i 1140 . . . 4 + ∈ V
138simp2i 1141 . . . 4 · ∈ V
1412, 13op1st 7933 . . 3 (1st ‘⟨ + , · ⟩) = +
152, 11, 143eqtrri 2766 . 2 + = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6849 . 2 ( +𝑣𝑈) = ( +𝑣 ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2764 1 + = ( +𝑣𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3447  cop 4596  cfv 6500  1st c1st 7923  NrmCVeccnv 29575   +𝑣 cpv 29576   + cva 29911   · csm 29912  normcno 29914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-oprab 7365  df-1st 7925  df-vc 29550  df-nv 29583  df-va 29586
This theorem is referenced by:  h2hvs  29968  axhfvadd-zf  29973  axhvcom-zf  29974  axhvass-zf  29975  axhvaddid-zf  29977  axhvdistr1-zf  29981  axhvdistr2-zf  29982  axhis2-zf  29986  hhva  30157
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