Theorem tcphval 25116

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	⊢ , = (·𝑖‘𝑊)

Assertion

Ref	Expression
tcphval	⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(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.

Step	Hyp	Ref	Expression
1		tcphval.n	. 2 ⊢ 𝐺 = (toℂPreHil‘𝑊)
2		id 22	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
3		fveq2 6822	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		tcphval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2782	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6		fveq2 6822	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊))
7		tcphval.h	. . . . . . . . 9 ⊢ , = (·𝑖‘𝑊)
8	6, 7	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , )
9	8	oveqd 7366	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥))
10	9	fveq2d 6826	. . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
11	5, 10	mpteq12dv 5179	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
12	2, 11	oveq12d 7367	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13		df-tcph 25067	. . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))))
14		ovex 7382	. . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
15	12, 13, 14	fvmpt 6930	. . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16		fvprc 6814	. . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅)
17		reldmtng 24524	. . . . 5 ⊢ Rel dom toNrmGrp
18	17	ovprc1 7388	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
19	16, 18	eqtr4d 2767	. . 3 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
20	15, 19	pm2.61i 182	. 2 ⊢ (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
21	1, 20	eqtri 2752	1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173  ‘cfv 6482  (class class class)co 7349  √csqrt 15140  Basecbs 17120  ·_𝑖cip 17166   toNrmGrp ctng 24464  toℂPreHilctcph 25065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-tng 24470  df-tcph 25067

This theorem is referenced by:  tcphbas  25117  tchplusg  25118  tcphmulr  25120  tcphsca  25121  tcphvsca  25122  tcphip  25123  tcphtopn  25124  tchnmfval  25126  tcphds  25129  tcphcph  25135  rrxsca  25294  rrx0  25295  rrxdim  33581