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Theorem h2hsm 28133
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 2752 . . . 4 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
21smfval 27761 . . 3 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5073 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2828 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 27767 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1135 . . . . 5 norm ∈ V
103, 9op1st 7333 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6347 . . 3 (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (2nd ‘⟨ + , · ⟩)
128simp1i 1133 . . . 4 + ∈ V
138simp2i 1134 . . . 4 · ∈ V
1412, 13op2nd 7334 . . 3 (2nd ‘⟨ + , · ⟩) = ·
152, 11, 143eqtrri 2779 . 2 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6347 . 2 ( ·𝑠OLD𝑈) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2777 1 · = ( ·𝑠OLD𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1072   = wceq 1624  wcel 2131  Vcvv 3332  cop 4319  cfv 6041  1st c1st 7323  2nd c2nd 7324  NrmCVeccnv 27740   ·𝑠OLD cns 27743   + cva 28078   · csm 28079  normcno 28081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fo 6047  df-fv 6049  df-oprab 6809  df-1st 7325  df-2nd 7326  df-vc 27715  df-nv 27748  df-sm 27753
This theorem is referenced by:  h2hvs  28135  axhfvmul-zf  28145  axhvmulid-zf  28146  axhvmulass-zf  28147  axhvdistr1-zf  28148  axhvdistr2-zf  28149  axhvmul0-zf  28150  axhis3-zf  28154  hhsm  28327
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