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Theorem h2hsm 31004
Description: The scalar product 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
h2hsm · = ( ·𝑠OLD𝑈)

Proof of Theorem h2hsm
StepHypRef Expression
1 eqid 2735 . . . 4 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
21smfval 30634 . . 3 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5475 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2836 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 30640 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1140 . . . . 5 norm ∈ V
103, 9op1st 8021 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6910 . . 3 (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (2nd ‘⟨ + , · ⟩)
128simp1i 1138 . . . 4 + ∈ V
138simp2i 1139 . . . 4 · ∈ V
1412, 13op2nd 8022 . . 3 (2nd ‘⟨ + , · ⟩) = ·
152, 11, 143eqtrri 2768 . 2 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6910 . 2 ( ·𝑠OLD𝑈) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2766 1 · = ( ·𝑠OLD𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1086   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637  cfv 6563  1st c1st 8011  2nd c2nd 8012  NrmCVeccnv 30613   ·𝑠OLD cns 30616   + cva 30949   · csm 30950  normcno 30952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-sm 30626
This theorem is referenced by:  h2hvs  31006  axhfvmul-zf  31016  axhvmulid-zf  31017  axhvmulass-zf  31018  axhvdistr1-zf  31019  axhvdistr2-zf  31020  axhvmul0-zf  31021  axhis3-zf  31025  hhsm  31198
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