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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 2818 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | phlsrng.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2818 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2818 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2818 | . . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | eqid 2818 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20700 | . 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 1138 | 1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ↦ cmpt 5137 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 *𝑟cstv 16555 Scalarcsca 16556 ·𝑖cip 16558 0gc0g 16701 *-Ringcsr 19544 LMHom clmhm 19720 LVecclvec 19803 ringLModcrglmod 19870 PreHilcphl 20696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-iota 6307 df-fv 6356 df-ov 7148 df-phl 20698 |
This theorem is referenced by: iporthcom 20707 ip0r 20709 ipdi 20712 ip2di 20713 ipassr 20718 ipassr2 20719 phlssphl 20731 cphcjcl 23714 |
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