Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscph Structured version   Visualization version   GIF version

Theorem iscph 22778
 Description: A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complex numbers, with a norm defined. (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 3758 . . . . 5 (𝑊 ∈ (PreHil ∩ NrmMod) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod))
21anbi1i 727 . . . 4 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
3 df-3an 1033 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
42, 3bitr4i 266 . . 3 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
54anbi1i 727 . 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 fvex 6113 . . . . . 6 (Scalar‘𝑤) ∈ V
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8 fvex 6113 . . . . . . 7 (Base‘𝑓) ∈ V
98a1i 11 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
10 simplr 788 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = (Scalar‘𝑤))
11 simpll 786 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑤 = 𝑊)
1211fveq2d 6107 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = (Scalar‘𝑊))
13 iscph.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1412, 13syl6eqr 2662 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = 𝐹)
1510, 14eqtrd 2644 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
16 simpr 476 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = (Base‘𝑓))
1715fveq2d 6107 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = (Base‘𝐹))
18 iscph.k . . . . . . . . . . . 12 𝐾 = (Base‘𝐹)
1917, 18syl6eqr 2662 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = 𝐾)
2016, 19eqtrd 2644 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
2120oveq2d 6565 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
2215, 21eqeq12d 2625 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
2320ineq1d 3775 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∩ (0[,)+∞)) = (𝐾 ∩ (0[,)+∞)))
2423imaeq2d 5385 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√ “ (𝑘 ∩ (0[,)+∞))) = (√ “ (𝐾 ∩ (0[,)+∞))))
2524, 20sseq12d 3597 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ↔ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾))
2611fveq2d 6107 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = (norm‘𝑊))
27 iscph.n . . . . . . . . . 10 𝑁 = (norm‘𝑊)
2826, 27syl6eqr 2662 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = 𝑁)
2911fveq2d 6107 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = (Base‘𝑊))
30 iscph.v . . . . . . . . . . 11 𝑉 = (Base‘𝑊)
3129, 30syl6eqr 2662 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = 𝑉)
3211fveq2d 6107 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = (·𝑖𝑊))
33 iscph.h . . . . . . . . . . . . 13 , = (·𝑖𝑊)
3432, 33syl6eqr 2662 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = , )
3534oveqd 6566 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
3635fveq2d 6107 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
3731, 36mpteq12dv 4663 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
3828, 37eqeq12d 2625 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) ↔ 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
3922, 25, 383anbi123d 1391 . . . . . . 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))))))
40 3anass 1035 . . . . . . 7 ((𝐹 = (ℂflds 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
4139, 40syl6bb 275 . . . . . 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)))))))
429, 41sbcied 3439 . . . . 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)))))))
437, 42sbcied 3439 . . . 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)))))))
44 df-cph 22776 . . . 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))))}
4543, 44elrab2 3333 . . 3 (𝑊 ∈ ℂPreHil ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
46 anass 679 . . 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)))))))
4745, 46bitr4i 266 . 2 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
48 3anass 1035 . 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))))))
495, 47, 483bitr4i 291 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   “ cima 5041  ‘cfv 5804  (class class class)co 6549  0cc0 9815  +∞cpnf 9950  [,)cico 12048  √csqrt 13821  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771  ·𝑖cip 15773  ℂfldccnfld 19567  PreHilcphl 19788  normcnm 22191  NrmModcnlm 22195  ℂPreHilccph 22774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-cph 22776 This theorem is referenced by:  cphphl  22779  cphnlm  22780  cphsca  22787  cphsqrtcl  22792  cphnmfval  22800  tchcph  22844
 Copyright terms: Public domain W3C validator