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Theorem tcphval 23801
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 6643 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4 tcphval.v . . . . . . 7 𝑉 = (Base‘𝑊)
53, 4syl6eqr 2874 . . . . . 6 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6 fveq2 6643 . . . . . . . . 9 (𝑤 = 𝑊 → (·𝑖𝑤) = (·𝑖𝑊))
7 tcphval.h . . . . . . . . 9 , = (·𝑖𝑊)
86, 7syl6eqr 2874 . . . . . . . 8 (𝑤 = 𝑊 → (·𝑖𝑤) = , )
98oveqd 7147 . . . . . . 7 (𝑤 = 𝑊 → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
109fveq2d 6647 . . . . . 6 (𝑤 = 𝑊 → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
115, 10mpteq12dv 5124 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
122, 11oveq12d 7148 . . . 4 (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13 df-tcph 23753 . . . 4 toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
14 ovex 7163 . . . 4 (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
1512, 13, 14fvmpt 6741 . . 3 (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16 fvprc 6636 . . . 4 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅)
17 reldmtng 23223 . . . . 5 Rel dom toNrmGrp
1817ovprc1 7169 . . . 4 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
1916, 18eqtr4d 2859 . . 3 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
2015, 19pm2.61i 185 . 2 (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
211, 20eqtri 2844 1 𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2115  Vcvv 3471  c0 4266  cmpt 5119  cfv 6328  (class class class)co 7130  csqrt 14571  Basecbs 16462  ·𝑖cip 16549   toNrmGrp ctng 23164  toℂPreHilctcph 23751
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-tng 23170  df-tcph 23753
This theorem is referenced by:  tcphbas  23802  tchplusg  23803  tcphmulr  23805  tcphsca  23806  tcphvsca  23807  tcphip  23808  tcphtopn  23809  tchnmfval  23811  tcphds  23814  tcphcph  23820  rrxsca  23979  rrx0  23980  rrxdim  31023
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