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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 2732 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2732 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | 1, 2, 3, 4, 5 | iscph 24694 | . 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 1147 | 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 1087 = wceq 1541 β wcel 2106 β© cin 3947 β wss 3948 β¦ cmpt 5231 β cima 5679 βcfv 6543 (class class class)co 7411 0cc0 11112 +βcpnf 11247 [,)cico 13328 βcsqrt 15182 Basecbs 17146 βΎs cress 17175 Scalarcsca 17202 Β·πcip 17204 βfldccnfld 20950 PreHilcphl 21183 normcnm 24092 NrmModcnlm 24096 βPreHilccph 24690 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-rab 3433 df-v 3476 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 7414 df-cph 24692 |
This theorem is referenced by: cphnm 24717 cphnmf 24719 cphtcphnm 24754 cphsscph 24775 |
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