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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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