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Theorem h2hsm 28733
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 2820 . . . 4 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
21smfval 28363 . . 3 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5328 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2908 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 28369 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1137 . . . . 5 norm ∈ V
103, 9op1st 7671 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6645 . . 3 (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (2nd ‘⟨ + , · ⟩)
128simp1i 1135 . . . 4 + ∈ V
138simp2i 1136 . . . 4 · ∈ V
1412, 13op2nd 7672 . . 3 (2nd ‘⟨ + , · ⟩) = ·
152, 11, 143eqtrri 2848 . 2 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6645 . 2 ( ·𝑠OLD𝑈) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2846 1 · = ( ·𝑠OLD𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1083   = wceq 1537  wcel 2114  Vcvv 3470  cop 4545  cfv 6327  1st c1st 7661  2nd c2nd 7662  NrmCVeccnv 28342   ·𝑠OLD cns 28345   + cva 28678   · csm 28679  normcno 28681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fo 6333  df-fv 6335  df-oprab 7133  df-1st 7663  df-2nd 7664  df-vc 28317  df-nv 28350  df-sm 28355
This theorem is referenced by:  h2hvs  28735  axhfvmul-zf  28745  axhvmulid-zf  28746  axhvmulass-zf  28747  axhvdistr1-zf  28748  axhvdistr2-zf  28749  axhvmul0-zf  28750  axhis3-zf  28754  hhsm  28927
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