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Theorem h2hsm 30228
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 2733 . . . 4 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
21smfval 29858 . . 3 ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩) = (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩))
3 opex 5465 . . . . 5 ⟨ + , · ⟩ ∈ V
4 h2h.1 . . . . . . . 8 𝑈 = ⟨⟨ + , · ⟩, norm
5 h2h.2 . . . . . . . 8 𝑈 ∈ NrmCVec
64, 5eqeltrri 2831 . . . . . . 7 ⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec
7 nvex 29864 . . . . . . 7 (⟨⟨ + , · ⟩, norm⟩ ∈ NrmCVec → ( + ∈ V ∧ · ∈ V ∧ norm ∈ V))
86, 7ax-mp 5 . . . . . 6 ( + ∈ V ∧ · ∈ V ∧ norm ∈ V)
98simp3i 1142 . . . . 5 norm ∈ V
103, 9op1st 7983 . . . 4 (1st ‘⟨⟨ + , · ⟩, norm⟩) = ⟨ + , ·
1110fveq2i 6895 . . 3 (2nd ‘(1st ‘⟨⟨ + , · ⟩, norm⟩)) = (2nd ‘⟨ + , · ⟩)
128simp1i 1140 . . . 4 + ∈ V
138simp2i 1141 . . . 4 · ∈ V
1412, 13op2nd 7984 . . 3 (2nd ‘⟨ + , · ⟩) = ·
152, 11, 143eqtrri 2766 . 2 · = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
164fveq2i 6895 . 2 ( ·𝑠OLD𝑈) = ( ·𝑠OLD ‘⟨⟨ + , · ⟩, norm⟩)
1715, 16eqtr4i 2764 1 · = ( ·𝑠OLD𝑈)
Colors of variables: wff setvar class
Syntax hints:  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3475  cop 4635  cfv 6544  1st c1st 7973  2nd c2nd 7974  NrmCVeccnv 29837   ·𝑠OLD cns 29840   + cva 30173   · csm 30174  normcno 30176
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29812  df-nv 29845  df-sm 29850
This theorem is referenced by:  h2hvs  30230  axhfvmul-zf  30240  axhvmulid-zf  30241  axhvmulass-zf  30242  axhvdistr1-zf  30243  axhvdistr2-zf  30244  axhvmul0-zf  30245  axhis3-zf  30249  hhsm  30422
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