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Theorem tchval 23011
 Description: Define a function to augment a subcomplex 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 6189 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 tchval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2673 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6189 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖𝑤) = (·𝑖𝑊))
7 tchval.h . . . . . . . . 9 , = (·𝑖𝑊)
86, 7syl6eqr 2673 . . . . . . . 8 (𝑤 = 𝑊 → (·𝑖𝑤) = , )
98oveqd 6664 . . . . . . 7 (𝑤 = 𝑊 → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
109fveq2d 6193 . . . . . 6 (𝑤 = 𝑊 → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
115, 10mpteq12dv 4731 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
122, 11oveq12d 6665 . . . 4 (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13 df-tch 22963 . . . 4 toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
14 ovex 6675 . . . 4 (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
1512, 13, 14fvmpt 6280 . . 3 (𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16 fvprc 6183 . . . 4 𝑊 ∈ V → (toℂHil‘𝑊) = ∅)
17 reldmtng 22436 . . . . 5 Rel dom toNrmGrp
1817ovprc1 6681 . . . 4 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
1916, 18eqtr4d 2658 . . 3 𝑊 ∈ V → (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
2015, 19pm2.61i 176 . 2 (toℂHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
211, 20eqtri 2643 1 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1482   ∈ wcel 1989  Vcvv 3198  ∅c0 3913   ↦ cmpt 4727  ‘cfv 5886  (class class class)co 6647  √csqrt 13967  Basecbs 15851  ·𝑖cip 15940   toNrmGrp ctng 22377  toℂHilctch 22961 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-tng 22383  df-tch 22963 This theorem is referenced by:  tchbas  23012  tchplusg  23013  tchmulr  23015  tchsca  23016  tchvsca  23017  tchip  23018  tchtopn  23019  tchnmfval  23021  tchds  23024  tchcph  23030
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