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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 2610 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | phlsrng.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2610 | . . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
4 | eqid 2610 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2610 | . . 3 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | eqid 2610 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 19792 | . 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 1070 | 1 ⊢ (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 *𝑟cstv 15770 Scalarcsca 15771 ·𝑖cip 15773 0gc0g 15923 *-Ringcsr 18667 LMHom clmhm 18840 LVecclvec 18923 ringLModcrglmod 18990 PreHilcphl 19788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-iota 5768 df-fv 5812 df-ov 6552 df-phl 19790 |
This theorem is referenced by: iporthcom 19799 ip0r 19801 ipdi 19804 ip2di 19805 ipassr 19810 ipassr2 19811 cphcjcl 22791 |
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