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Theorem tcphval 25234
Description: Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
tcphval.n 𝐺 = (toℂPreHil‘𝑊)
tcphval.v 𝑉 = (Base‘𝑊)
tcphval.h , = (·𝑖𝑊)
Assertion
Ref Expression
tcphval 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Distinct variable groups:   𝑥, ,   𝑥,𝐺   𝑥,𝑉   𝑥,𝑊

Proof of Theorem tcphval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 tcphval.n . 2 𝐺 = (toℂPreHil‘𝑊)
2 id 22 . . . . 5 (𝑤 = 𝑊𝑤 = 𝑊)
3 fveq2 6893 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 tcphval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4eqtr4di 2784 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6893 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖𝑤) = (·𝑖𝑊))
7 tcphval.h . . . . . . . . 9 , = (·𝑖𝑊)
86, 7eqtr4di 2784 . . . . . . . 8 (𝑤 = 𝑊 → (·𝑖𝑤) = , )
98oveqd 7433 . . . . . . 7 (𝑤 = 𝑊 → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
109fveq2d 6897 . . . . . 6 (𝑤 = 𝑊 → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
115, 10mpteq12dv 5236 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
122, 11oveq12d 7434 . . . 4 (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13 df-tcph 25185 . . . 4 toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
14 ovex 7449 . . . 4 (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
1512, 13, 14fvmpt 7001 . . 3 (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16 fvprc 6885 . . . 4 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅)
17 reldmtng 24635 . . . . 5 Rel dom toNrmGrp
1817ovprc1 7455 . . . 4 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
1916, 18eqtr4d 2769 . . 3 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
2015, 19pm2.61i 182 . 2 (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
211, 20eqtri 2754 1 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3462  c0 4322  cmpt 5228  cfv 6546  (class class class)co 7416  csqrt 15233  Basecbs 17208  ·𝑖cip 17266   toNrmGrp ctng 24575  toℂPreHilctcph 25183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-iota 6498  df-fun 6548  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-tng 24581  df-tcph 25185
This theorem is referenced by:  tcphbas  25235  tchplusg  25236  tcphmulr  25238  tcphsca  25239  tcphvsca  25240  tcphip  25241  tcphtopn  25242  tchnmfval  25244  tcphds  25247  tcphcph  25253  rrxsca  25412  rrx0  25413  rrxdim  33515
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