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Mirrors > Home > MPE Home > Th. List > phlsrng | Structured version Visualization version GIF version |
Description: The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
phlsrng | ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | phlsrng.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2736 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2736 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2736 | . . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | eqid 2736 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 21028 | . 2 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖‘𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘𝐹)‘(𝑥(·𝑖‘𝑊)𝑦)) = (𝑦(·𝑖‘𝑊)𝑥)))) |
8 | 7 | simp2bi 1146 | 1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3063 ↦ cmpt 5187 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 *𝑟cstv 17132 Scalarcsca 17133 ·𝑖cip 17135 0gc0g 17318 *-Ringcsr 20299 LMHom clmhm 20476 LVecclvec 20559 ringLModcrglmod 20626 PreHilcphl 21024 |
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 5262 |
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 2943 df-ral 3064 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-iota 6446 df-fv 6502 df-ov 7357 df-phl 21026 |
This theorem is referenced by: iporthcom 21035 ip0r 21037 ipdi 21040 ip2di 21041 ipassr 21046 ipassr2 21047 phlssphl 21059 cphcjcl 24543 |
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