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Theorem iscph 22878
Description: A subcomplex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the field of 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 3774 . . . . 5 (𝑊 ∈ (PreHil ∩ NrmMod) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod))
21anbi1i 730 . . . 4 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
3 df-3an 1038 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)))
42, 3bitr4i 267 . . 3 ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ↔ (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
54anbi1i 730 . 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 6158 . . . . . 6 (Scalar‘𝑤) ∈ V
76a1i 11 . . . . 5 (𝑤 = 𝑊 → (Scalar‘𝑤) ∈ V)
8 fvex 6158 . . . . . . 7 (Base‘𝑓) ∈ V
98a1i 11 . . . . . 6 ((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) → (Base‘𝑓) ∈ V)
10 simplr 791 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = (Scalar‘𝑤))
11 simpll 789 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑤 = 𝑊)
1211fveq2d 6152 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = (Scalar‘𝑊))
13 iscph.f . . . . . . . . . . 11 𝐹 = (Scalar‘𝑊)
1412, 13syl6eqr 2673 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Scalar‘𝑤) = 𝐹)
1510, 14eqtrd 2655 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑓 = 𝐹)
16 simpr 477 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = (Base‘𝑓))
1715fveq2d 6152 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = (Base‘𝐹))
18 iscph.k . . . . . . . . . . . 12 𝐾 = (Base‘𝐹)
1917, 18syl6eqr 2673 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑓) = 𝐾)
2016, 19eqtrd 2655 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → 𝑘 = 𝐾)
2120oveq2d 6620 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (ℂflds 𝑘) = (ℂflds 𝐾))
2215, 21eqeq12d 2636 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑓 = (ℂflds 𝑘) ↔ 𝐹 = (ℂflds 𝐾)))
2320ineq1d 3791 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑘 ∩ (0[,)+∞)) = (𝐾 ∩ (0[,)+∞)))
2423imaeq2d 5425 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√ “ (𝑘 ∩ (0[,)+∞))) = (√ “ (𝐾 ∩ (0[,)+∞))))
2524, 20sseq12d 3613 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ↔ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾))
2611fveq2d 6152 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = (norm‘𝑊))
27 iscph.n . . . . . . . . . 10 𝑁 = (norm‘𝑊)
2826, 27syl6eqr 2673 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (norm‘𝑤) = 𝑁)
2911fveq2d 6152 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = (Base‘𝑊))
30 iscph.v . . . . . . . . . . 11 𝑉 = (Base‘𝑊)
3129, 30syl6eqr 2673 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (Base‘𝑤) = 𝑉)
3211fveq2d 6152 . . . . . . . . . . . . 13 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = (·𝑖𝑊))
33 iscph.h . . . . . . . . . . . . 13 , = (·𝑖𝑊)
3432, 33syl6eqr 2673 . . . . . . . . . . . 12 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (·𝑖𝑤) = , )
3534oveqd 6621 . . . . . . . . . . 11 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥(·𝑖𝑤)𝑥) = (𝑥 , 𝑥))
3635fveq2d 6152 . . . . . . . . . 10 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (√‘(𝑥(·𝑖𝑤)𝑥)) = (√‘(𝑥 , 𝑥)))
3731, 36mpteq12dv 4693 . . . . . . . . 9 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
3828, 37eqeq12d 2636 . . . . . . . 8 (((𝑤 = 𝑊𝑓 = (Scalar‘𝑤)) ∧ 𝑘 = (Base‘𝑓)) → ((norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))) ↔ 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
3922, 25, 383anbi123d 1396 . . . . . . 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 1040 . . . . . . 7 ((𝐹 = (ℂflds 𝐾) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))) ↔ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
4139, 40syl6bb 276 . . . . . 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 3454 . . . . 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 3454 . . . 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 22876 . . . 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 3348 . . 3 (𝑊 ∈ ℂPreHil ↔ (𝑊 ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (ℂflds 𝐾) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))))
46 anass 680 . . 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 267 . 2 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (ℂflds 𝐾)) ∧ ((√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))))
48 3anass 1040 . 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 292 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3186  [wsbc 3417  cin 3554  wss 3555  cmpt 4673  cima 5077  cfv 5847  (class class class)co 6604  0cc0 9880  +∞cpnf 10015  [,)cico 12119  csqrt 13907  Basecbs 15781  s cress 15782  Scalarcsca 15865  ·𝑖cip 15867  fldccnfld 19665  PreHilcphl 19888  normcnm 22291  NrmModcnlm 22295  ℂPreHilccph 22874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607  df-cph 22876
This theorem is referenced by:  cphphl  22879  cphnlm  22880  cphsca  22887  cphsqrtcl  22892  cphnmfval  22900  tchcph  22944
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