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Theorem iscph 24687
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 = (Baseβ€˜π‘Š)
iscph.h , = (Β·π‘–β€˜π‘Š)
iscph.n 𝑁 = (normβ€˜π‘Š)
iscph.f 𝐹 = (Scalarβ€˜π‘Š)
iscph.k 𝐾 = (Baseβ€˜πΉ)
Assertion
Ref Expression
iscph (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
Distinct variable group:   π‘₯,π‘Š
Allowed substitution hints:   𝐹(π‘₯)   , (π‘₯)   𝐾(π‘₯)   𝑁(π‘₯)   𝑉(π‘₯)

Proof of Theorem iscph
Dummy variables 𝑓 π‘˜ 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elin 3965 . . . . 5 (π‘Š ∈ (PreHil ∩ NrmMod) ↔ (π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod))
21anbi1i 625 . . . 4 ((π‘Š ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)))
3 df-3an 1090 . . . 4 ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)))
42, 3bitr4i 278 . . 3 ((π‘Š ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ↔ (π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)))
54anbi1i 625 . 2 (((π‘Š ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))) ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))))
6 fvexd 6907 . . . . 5 (𝑀 = π‘Š β†’ (Scalarβ€˜π‘€) ∈ V)
7 fvexd 6907 . . . . . 6 ((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) β†’ (Baseβ€˜π‘“) ∈ V)
8 simplr 768 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ 𝑓 = (Scalarβ€˜π‘€))
9 simpll 766 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ 𝑀 = π‘Š)
109fveq2d 6896 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Scalarβ€˜π‘€) = (Scalarβ€˜π‘Š))
11 iscph.f . . . . . . . . . . 11 𝐹 = (Scalarβ€˜π‘Š)
1210, 11eqtr4di 2791 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Scalarβ€˜π‘€) = 𝐹)
138, 12eqtrd 2773 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ 𝑓 = 𝐹)
14 simpr 486 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ π‘˜ = (Baseβ€˜π‘“))
1513fveq2d 6896 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Baseβ€˜π‘“) = (Baseβ€˜πΉ))
16 iscph.k . . . . . . . . . . . 12 𝐾 = (Baseβ€˜πΉ)
1715, 16eqtr4di 2791 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Baseβ€˜π‘“) = 𝐾)
1814, 17eqtrd 2773 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ π‘˜ = 𝐾)
1918oveq2d 7425 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (β„‚fld β†Ύs π‘˜) = (β„‚fld β†Ύs 𝐾))
2013, 19eqeq12d 2749 . . . . . . . 8 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (𝑓 = (β„‚fld β†Ύs π‘˜) ↔ 𝐹 = (β„‚fld β†Ύs 𝐾)))
2118ineq1d 4212 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (π‘˜ ∩ (0[,)+∞)) = (𝐾 ∩ (0[,)+∞)))
2221imaeq2d 6060 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (√ β€œ (π‘˜ ∩ (0[,)+∞))) = (√ β€œ (𝐾 ∩ (0[,)+∞))))
2322, 18sseq12d 4016 . . . . . . . 8 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ ((√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ↔ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾))
249fveq2d 6896 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (normβ€˜π‘€) = (normβ€˜π‘Š))
25 iscph.n . . . . . . . . . 10 𝑁 = (normβ€˜π‘Š)
2624, 25eqtr4di 2791 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (normβ€˜π‘€) = 𝑁)
279fveq2d 6896 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
28 iscph.v . . . . . . . . . . 11 𝑉 = (Baseβ€˜π‘Š)
2927, 28eqtr4di 2791 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Baseβ€˜π‘€) = 𝑉)
309fveq2d 6896 . . . . . . . . . . . . 13 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Β·π‘–β€˜π‘€) = (Β·π‘–β€˜π‘Š))
31 iscph.h . . . . . . . . . . . . 13 , = (Β·π‘–β€˜π‘Š)
3230, 31eqtr4di 2791 . . . . . . . . . . . 12 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (Β·π‘–β€˜π‘€) = , )
3332oveqd 7426 . . . . . . . . . . 11 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (π‘₯(Β·π‘–β€˜π‘€)π‘₯) = (π‘₯ , π‘₯))
3433fveq2d 6896 . . . . . . . . . 10 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)) = (βˆšβ€˜(π‘₯ , π‘₯)))
3529, 34mpteq12dv 5240 . . . . . . . . 9 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))) = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
3626, 35eqeq12d 2749 . . . . . . . 8 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ ((normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))) ↔ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
3720, 23, 363anbi123d 1437 . . . . . . 7 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ ((𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))) ↔ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))))
38 3anass 1096 . . . . . . 7 ((𝐹 = (β„‚fld β†Ύs 𝐾) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))) ↔ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))))
3937, 38bitrdi 287 . . . . . 6 (((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) ∧ π‘˜ = (Baseβ€˜π‘“)) β†’ ((𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))) ↔ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))))
407, 39sbcied 3823 . . . . 5 ((𝑀 = π‘Š ∧ 𝑓 = (Scalarβ€˜π‘€)) β†’ ([(Baseβ€˜π‘“) / π‘˜](𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))) ↔ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))))
416, 40sbcied 3823 . . . 4 (𝑀 = π‘Š β†’ ([(Scalarβ€˜π‘€) / 𝑓][(Baseβ€˜π‘“) / π‘˜](𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))) ↔ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))))
42 df-cph 24685 . . . 4 β„‚PreHil = {𝑀 ∈ (PreHil ∩ NrmMod) ∣ [(Scalarβ€˜π‘€) / 𝑓][(Baseβ€˜π‘“) / π‘˜](𝑓 = (β„‚fld β†Ύs π‘˜) ∧ (√ β€œ (π‘˜ ∩ (0[,)+∞))) βŠ† π‘˜ ∧ (normβ€˜π‘€) = (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))))}
4341, 42elrab2 3687 . . 3 (π‘Š ∈ β„‚PreHil ↔ (π‘Š ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))))
44 anass 470 . . 3 (((π‘Š ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))) ↔ (π‘Š ∈ (PreHil ∩ NrmMod) ∧ (𝐹 = (β„‚fld β†Ύs 𝐾) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))))
4543, 44bitr4i 278 . 2 (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ (PreHil ∩ NrmMod) ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))))
46 3anass 1096 . 2 (((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))) ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ ((√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))))
475, 45, 463bitr4i 303 1 (π‘Š ∈ β„‚PreHil ↔ ((π‘Š ∈ PreHil ∧ π‘Š ∈ NrmMod ∧ 𝐹 = (β„‚fld β†Ύs 𝐾)) ∧ (√ β€œ (𝐾 ∩ (0[,)+∞))) βŠ† 𝐾 ∧ 𝑁 = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475  [wsbc 3778   ∩ cin 3948   βŠ† wss 3949   ↦ cmpt 5232   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  +∞cpnf 11245  [,)cico 13326  βˆšcsqrt 15180  Basecbs 17144   β†Ύs cress 17173  Scalarcsca 17200  Β·π‘–cip 17202  β„‚fldccnfld 20944  PreHilcphl 21177  normcnm 24085  NrmModcnlm 24089  β„‚PreHilccph 24683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-cph 24685
This theorem is referenced by:  cphphl  24688  cphnlm  24689  cphsca  24696  cphsqrtcl  24701  cphnmfval  24709  tcphcph  24754  cphsscph  24768
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