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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 2736 | . . 3 β’ (Scalarβπ) = (Scalarβπ) | |
5 | eqid 2736 | . . 3 β’ (Baseβ(Scalarβπ)) = (Baseβ(Scalarβπ)) | |
6 | 1, 2, 3, 4, 5 | iscph 24440 | . 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 3897 β wss 3898 β¦ cmpt 5175 β cima 5623 βcfv 6479 (class class class)co 7337 0cc0 10972 +βcpnf 11107 [,)cico 13182 βcsqrt 15043 Basecbs 17009 βΎs cress 17038 Scalarcsca 17062 Β·πcip 17064 βfldccnfld 20703 PreHilcphl 20935 normcnm 23838 NrmModcnlm 23842 βPreHilccph 24436 |
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 2707 ax-nul 5250 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-rab 3404 df-v 3443 df-sbc 3728 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-br 5093 df-opab 5155 df-mpt 5176 df-xp 5626 df-cnv 5628 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fv 6487 df-ov 7340 df-cph 24438 |
This theorem is referenced by: cphnm 24463 cphnmf 24465 cphtcphnm 24500 cphsscph 24521 |
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