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Theorem iscph 23775
 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 , = (·𝑖𝑊)
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 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(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.
StepHypRef Expression
1 elin 3897 . . . . 5 (𝑊 ∈ (PreHil ∩ NrmMod) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod))
21anbi1i 626 . . . 4 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
3 df-3an 1086 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
42, 3bitr4i 281 . . 3 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
54anbi1i 626 . 2 (((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
6 fvexd 6660 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
7 fvexd 6660 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
8 simplr 768 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = (Scalar‘𝑤))
9 simpll 766 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑤 = 𝑊)
109fveq2d 6649 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = (Scalar‘𝑊))
11 iscph.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1210, 11eqtr4di 2851 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = 𝐹)
138, 12eqtrd 2833 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
14 simpr 488 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = (Base‘𝑓))
1513fveq2d 6649 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = (Base‘𝐹))
16 iscph.k . . . . . . . . . . . 12 𝐾 = (Base‘𝐹)
1715, 16eqtr4di 2851 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = 𝐾)
1814, 17eqtrd 2833 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
1918oveq2d 7151 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
2013, 19eqeq12d 2814 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
2118ineq1d 4138 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∩ (0[,)+∞)) = (𝐾 ∩ (0[,)+∞)))
2221imaeq2d 5896 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√ “ (𝑘 ∩ (0[,)+∞))) = (√ “ (𝐾 ∩ (0[,)+∞))))
2322, 18sseq12d 3948 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ↔ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾))
249fveq2d 6649 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = (norm‘𝑊))
25 iscph.n . . . . . . . . . 10 𝑁 = (norm‘𝑊)
2624, 25eqtr4di 2851 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = 𝑁)
279fveq2d 6649 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = (Base‘𝑊))
28 iscph.v . . . . . . . . . . 11 𝑉 = (Base‘𝑊)
2927, 28eqtr4di 2851 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = 𝑉)
309fveq2d 6649 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = (·𝑖𝑊))
31 iscph.h . . . . . . . . . . . . 13 , = (·𝑖𝑊)
3230, 31eqtr4di 2851 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = , )
3332oveqd 7152 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
3433fveq2d 6649 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
3529, 34mpteq12dv 5115 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
3626, 35eqeq12d 2814 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) ↔ 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
3720, 23, 363anbi123d 1433 . . . . . . 7 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
38 3anass 1092 . . . . . . 7 ((𝐹 = (ℂflds 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
3937, 38syl6bb 290 . . . . . 6 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
407, 39sbcied 3762 . . . . 5 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → ([(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
416, 40sbcied 3762 . . . 4 (𝑤 = 𝑊 → ([(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
42 df-cph 23773 . . . 4 ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}
4341, 42elrab2 3631 . . 3 (𝑊 ∈ ℂPreHil ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
44 anass 472 . . 3 (((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))) ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
4543, 44bitr4i 281 . 2 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
46 3anass 1092 . 2 (((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
475, 45, 463bitr4i 306 1 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  Vcvv 3441  [wsbc 3720   ∩ cin 3880   ⊆ wss 3881   ↦ cmpt 5110   “ cima 5522  ‘cfv 6324  (class class class)co 7135  0cc0 10526  +∞cpnf 10661  [,)cico 12728  √csqrt 14584  Basecbs 16475   ↾s cress 16476  Scalarcsca 16560  ·𝑖cip 16562  ℂfldccnfld 20091  PreHilcphl 20313  normcnm 23183  NrmModcnlm 23187  ℂPreHilccph 23771 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-ov 7138  df-cph 23773 This theorem is referenced by:  cphphl  23776  cphnlm  23777  cphsca  23784  cphsqrtcl  23789  cphnmfval  23797  tcphcph  23841  cphsscph  23855
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