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Theorem tchval 22825
 Description: Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
tchval.n 𝐺 = (toℂHil‘𝑊)
tchval.v 𝑉 = (Base‘𝑊)
tchval.h , = (·𝑖𝑊)
Assertion
Ref Expression
tchval 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Distinct variable groups:   𝑥, ,   𝑥,𝐺   𝑥,𝑉   𝑥,𝑊

Proof of Theorem tchval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 tchval.n . 2 𝐺 = (toℂHil‘𝑊)
2 id 22 . . . . 5 (𝑤 = 𝑊𝑤 = 𝑊)
3 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 tchval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖𝑤) = (·𝑖𝑊))
7 tchval.h . . . . . . . . 9 , = (·𝑖𝑊)
86, 7syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → (·𝑖𝑤) = , )
98oveqd 6566 . . . . . . 7 (𝑤 = 𝑊 → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
109fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
115, 10mpteq12dv 4663 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
122, 11oveq12d 6567 . . . 4 (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13 df-tch 22777 . . . 4 toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
14 ovex 6577 . . . 4 (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
1512, 13, 14fvmpt 6191 . . 3 (𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16 fvprc 6097 . . . 4 𝑊 ∈ V → (toℂHil‘𝑊) = ∅)
17 reldmtng 22252 . . . . 5 Rel dom toNrmGrp
1817ovprc1 6582 . . . 4 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
1916, 18eqtr4d 2647 . . 3 𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
2015, 19pm2.61i 175 . 2 (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
211, 20eqtri 2632 1 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  √csqrt 13821  Basecbs 15695  ·𝑖cip 15773   toNrmGrp ctng 22193  toℂHilctch 22775 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-tng 22199  df-tch 22777 This theorem is referenced by:  tchbas  22826  tchplusg  22827  tchmulr  22829  tchsca  22830  tchvsca  22831  tchip  22832  tchtopn  22833  tchnmfval  22835  tchds  22838  tchcph  22844
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