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Mirrors > Home > MPE Home > Th. List > cphnlm | Structured version Visualization version GIF version |
Description: A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
cphnlm | ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2736 | . . . 4 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2736 | . . . 4 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
4 | eqid 2736 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
5 | eqid 2736 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
6 | 1, 2, 3, 4, 5 | iscph 24532 | . . 3 ⊢ (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖‘𝑊)𝑥))))) |
7 | 6 | simp1bi 1145 | . 2 ⊢ (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂfld ↾s (Base‘(Scalar‘𝑊))))) |
8 | 7 | simp2d 1143 | 1 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∩ cin 3909 ⊆ wss 3910 ↦ cmpt 5188 “ cima 5636 ‘cfv 6496 (class class class)co 7356 0cc0 11050 +∞cpnf 11185 [,)cico 13265 √csqrt 15117 Basecbs 17082 ↾s cress 17111 Scalarcsca 17135 ·𝑖cip 17137 ℂfldccnfld 20794 PreHilcphl 21026 normcnm 23930 NrmModcnlm 23934 ℂPreHilccph 24528 |
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 2707 ax-nul 5263 |
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 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-rab 3408 df-v 3447 df-sbc 3740 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-xp 5639 df-cnv 5641 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fv 6504 df-ov 7359 df-cph 24530 |
This theorem is referenced by: cphngp 24535 cphlmod 24536 cphnvc 24538 cphnmvs 24552 ipcnlem2 24606 ipcnlem1 24607 csscld 24611 cphsscph 24613 |
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