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Mirrors > Home > MPE Home > Th. List > cphnmfval | Structured version Visualization version GIF version |
Description: The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
nmsq.v | β’ π = (Baseβπ) |
nmsq.h | β’ , = (Β·πβπ) |
nmsq.n | β’ π = (normβπ) |
Ref | Expression |
---|---|
cphnmfval | β’ (π β βPreHil β π = (π₯ β π β¦ (ββ(π₯ , π₯)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmsq.v | . . 3 β’ π = (Baseβπ) | |
2 | nmsq.h | . . 3 β’ , = (Β·πβπ) | |
3 | nmsq.n | . . 3 β’ π = (normβπ) | |
4 | eqid 2731 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2731 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | 1, 2, 3, 4, 5 | iscph 24919 | . 2 β’ (π β βPreHil β ((π β PreHil β§ π β NrmMod β§ (Scalarβπ) = (βfld βΎs (Baseβ(Scalarβπ)))) β§ (β β ((Baseβ(Scalarβπ)) β© (0[,)+β))) β (Baseβ(Scalarβπ)) β§ π = (π₯ β π β¦ (ββ(π₯ , π₯))))) |
7 | 6 | simp3bi 1146 | 1 β’ (π β βPreHil β π = (π₯ β π β¦ (ββ(π₯ , π₯)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1086 = wceq 1540 β wcel 2105 β© cin 3947 β wss 3948 β¦ cmpt 5231 β cima 5679 βcfv 6543 (class class class)co 7412 0cc0 11114 +βcpnf 11250 [,)cico 13331 βcsqrt 15185 Basecbs 17149 βΎs cress 17178 Scalarcsca 17205 Β·πcip 17207 βfldccnfld 21145 PreHilcphl 21397 normcnm 24306 NrmModcnlm 24310 βPreHilccph 24915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fv 6551 df-ov 7415 df-cph 24917 |
This theorem is referenced by: cphnm 24942 cphnmf 24944 cphtcphnm 24979 cphsscph 25000 |
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