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Theorem h2hsm 30995
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 2736 . . . 4 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
21smfval 30625 . . 3 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5468 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2837 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 30631 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1141 . . . . 5 norm ∈ V
103, 9op1st 8023 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6908 . . 3 (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (2nd ‘⟨ + , · ⟩)
128simp1i 1139 . . . 4 + ∈ V
138simp2i 1140 . . . 4 · ∈ V
1412, 13op2nd 8024 . . 3 (2nd ‘⟨ + , · ⟩) = ·
152, 11, 143eqtrri 2769 . 2 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6908 . 2 ( ·𝑠OLD𝑈) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2767 1 · = ( ·𝑠OLD𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1086   = wceq 1539  wcel 2107  Vcvv 3479  cop 4631  cfv 6560  1st c1st 8013  2nd c2nd 8014  NrmCVeccnv 30604   ·𝑠OLD cns 30607   + cva 30940   · csm 30941  normcno 30943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-oprab 7436  df-1st 8015  df-2nd 8016  df-vc 30579  df-nv 30612  df-sm 30617
This theorem is referenced by:  h2hvs  30997  axhfvmul-zf  31007  axhvmulid-zf  31008  axhvmulass-zf  31009  axhvdistr1-zf  31010  axhvdistr2-zf  31011  axhvmul0-zf  31012  axhis3-zf  31016  hhsm  31189
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