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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 | ⊢ 𝑉 = (Base‘𝑊) |
nmsq.h | ⊢ , = (·𝑖‘𝑊) |
nmsq.n | ⊢ 𝑁 = (norm‘𝑊) |
Ref | Expression |
---|---|
cphnmfval | ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nmsq.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | nmsq.h | . . 3 ⊢ , = (·𝑖‘𝑊) | |
3 | nmsq.n | . . 3 ⊢ 𝑁 = (norm‘𝑊) | |
4 | eqid 2622 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2622 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 22970 | . 2 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))))) |
7 | 6 | simp3bi 1078 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ↦ cmpt 4729 “ cima 5117 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 [,)cico 12177 √csqrt 13973 Basecbs 15857 ↾s cress 15858 Scalarcsca 15944 ·𝑖cip 15946 ℂfldccnfld 19746 PreHilcphl 19969 normcnm 22381 NrmModcnlm 22385 ℂPreHilccph 22966 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-ov 6653 df-cph 22968 |
This theorem is referenced by: cphnm 22993 cphnmf 22995 cphtchnm 23029 |
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