Theorem iscph 25223

Description: A subcomplex pre-Hilbert space is exactly a pre-Hilbert space over a subfield of the field of complex numbers closed under square roots of nonnegative reals equipped with a norm. (Contributed by Mario Carneiro, 8-Oct-2015.)

Hypotheses

Ref	Expression
iscph.v	⊢ 𝑉 = (Base‘𝑊)
iscph.h	⊢ , = (·𝑖‘𝑊)
iscph.n	⊢ 𝑁 = (norm‘𝑊)
iscph.f	⊢ 𝐹 = (Scalar‘𝑊)
iscph.k	⊢ 𝐾 = (Base‘𝐹)

Assertion

Ref	Expression
iscph	⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))

Distinct variable group:   𝑥,𝑊

Allowed substitution hints:   𝐹(𝑥)   , (𝑥)   𝐾(𝑥)   𝑁(𝑥)   𝑉(𝑥)

Proof of Theorem iscph

Dummy variables 𝑓 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3992	. . . . 5 ⊢ (𝑊 ∈ (PreHil ∩ NrmMod) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod))
2	1	anbi1i 623	. . . 4 ⊢ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)))
3		df-3an 1089	. . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)))
4	2, 3	bitr4i 278	. . 3 ⊢ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)))
5	4	anbi1i 623	. 2 ⊢ (((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))))
6		fvexd 6935	. . . . 5 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
7		fvexd 6935	. . . . . 6 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
8		simplr 768	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = (Scalar‘𝑤))
9		simpll 766	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑤 = 𝑊)
10	9	fveq2d 6924	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = (Scalar‘𝑊))
11		iscph.f	. . . . . . . . . . 11 ⊢ 𝐹 = (Scalar‘𝑊)
12	10, 11	eqtr4di 2798	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = 𝐹)
13	8, 12	eqtrd 2780	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
14		simpr 484	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = (Base‘𝑓))
15	13	fveq2d 6924	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = (Base‘𝐹))
16		iscph.k	. . . . . . . . . . . 12 ⊢ 𝐾 = (Base‘𝐹)
17	15, 16	eqtr4di 2798	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = 𝐾)
18	14, 17	eqtrd 2780	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
19	18	oveq2d 7464	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂfld ↾s 𝑘) = (ℂfld ↾s 𝐾))
20	13, 19	eqeq12d 2756	. . . . . . . 8 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂfld ↾s 𝑘) ↔ 𝐹 = (ℂfld ↾s 𝐾)))
21	18	ineq1d 4240	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∩ (0[,)+∞)) = (𝐾 ∩ (0[,)+∞)))
22	21	imaeq2d 6089	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√ “ (𝑘 ∩ (0[,)+∞))) = (√ “ (𝐾 ∩ (0[,)+∞))))
23	22, 18	sseq12d 4042	. . . . . . . 8 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ↔ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾))
24	9	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = (norm‘𝑊))
25		iscph.n	. . . . . . . . . 10 ⊢ 𝑁 = (norm‘𝑊)
26	24, 25	eqtr4di 2798	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = 𝑁)
27	9	fveq2d 6924	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = (Base‘𝑊))
28		iscph.v	. . . . . . . . . . 11 ⊢ 𝑉 = (Base‘𝑊)
29	27, 28	eqtr4di 2798	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = 𝑉)
30	9	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖‘𝑤) = (·𝑖‘𝑊))
31		iscph.h	. . . . . . . . . . . . 13 ⊢ , = (·𝑖‘𝑊)
32	30, 31	eqtr4di 2798	. . . . . . . . . . . 12 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖‘𝑤) = , )
33	32	oveqd 7465	. . . . . . . . . . 11 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥(·𝑖‘𝑤)𝑥) = (𝑥 , 𝑥))
34	33	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√‘(𝑥(·𝑖‘𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
35	29, 34	mpteq12dv 5257	. . . . . . . . 9 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))
36	26, 35	eqeq12d 2756	. . . . . . . 8 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))) ↔ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))
37	20, 23, 36	3anbi123d 1436	. . . . . . 7 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))))
38		3anass 1095	. . . . . . 7 ⊢ ((𝐹 = (ℂfld ↾s 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))))
39	37, 38	bitrdi 287	. . . . . 6 ⊢ (((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
40	7, 39	sbcied 3850	. . . . 5 ⊢ ((𝑤 = 𝑊 ∧ 𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
41	6, 40	sbcied 3850	. . . 4 ⊢ (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥)))) ↔ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
42		df-cph 25221	. . . 4 ⊢ ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖‘𝑤)𝑥))))}
43	41, 42	elrab2 3711	. . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
44		anass 468	. . 3 ⊢ (((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂfld ↾s 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
45	43, 44	bitr4i 278	. 2 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))))
46		3anass 1095	. 2 ⊢ (((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))))
47	5, 45, 46	3bitr4i 303	1 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂfld ↾s 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  Vcvv 3488  [wsbc 3804   ∩ cin 3975   ⊆ wss 3976   ↦ cmpt 5249   “ cima 5703  ‘cfv 6573  (class class class)co 7448  0cc0 11184  +∞cpnf 11321  [,)cico 13409  √csqrt 15282  Basecbs 17258   ↾_s cress 17287  Scalarcsca 17314  ·_𝑖cip 17316  ℂ_fldccnfld 21387  PreHilcphl 21665  normcnm 24610  NrmModcnlm 24614  ℂPreHilccph 25219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-nul 5324

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fv 6581  df-ov 7451  df-cph 25221

This theorem is referenced by:  cphphl  25224  cphnlm  25225  cphsca  25232  cphsqrtcl  25237  cphnmfval  25245  tcphcph  25290  cphsscph  25304