Theorem tcphval 25358

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 23	. . . . 5 ⊢ (𝑤 = 𝑊 → 𝑤 = 𝑊)
3		fveq2 6883	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
4		tcphval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑊)
5	3, 4	eqtr4di 2816	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
6		fveq2 6883	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = (·𝑖‘𝑊))
7		tcphval.h	. . . . . . . . 9 ⊢ , = (·𝑖‘𝑊)
8	6, 7	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (·𝑖‘𝑤) = , )
9	8	oveqd 7429	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥))
10	9	fveq2d 6887	. . . . . 6 ⊢ (𝑤 = 𝑊 → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
11	5, 10	mpteq12dv 5199	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
12	2, 11	oveq12d 7430	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
13		df-tcph 25309	. . . 4 ⊢ toℂPreHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))))
14		ovex 7445	. . . 4 ⊢ (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ∈ V
15	12, 13, 14	fvmpt 6991	. . 3 ⊢ (𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
16		fvprc 6875	. . . 4 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = ∅)
17		reldmtng 24776	. . . . 5 ⊢ Rel dom toNrmGrp
18	17	ovprc1 7451	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) = ∅)
19	16, 18	eqtr4d 2801	. . 3 ⊢ (¬ 𝑊 ∈ V → (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
20	15, 19	pm2.61i 184	. 2 ⊢ (toℂPreHil‘𝑊) = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
21	1, 20	eqtri 2786	1 ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412  √csqrt 15286  Basecbs 17270  ·_𝑖cip 17316   toNrmGrp ctng 24716  toℂPreHilctcph 25307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-tng 24722  df-tcph 25309

This theorem is referenced by:  tcphbas  25359  tchplusg  25360  tcphmulr  25362  tcphsca  25363  tcphvsca  25364  tcphip  25365  tcphtopn  25366  tchnmfval  25368  tcphds  25371  tcphcph  25377  rrxsca  25536  rrx0  25537  rrxdim  33985