![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > phllmhm | Structured version Visualization version GIF version |
Description: The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
phllmhm.g | ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) |
Ref | Expression |
---|---|
phllmhm | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phllmhm.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
2 | phlsrng.f | . . . . 5 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | phllmhm.h | . . . . 5 ⊢ , = (·𝑖‘𝑊) | |
4 | eqid 2798 | . . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2798 | . . . . 5 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹) | |
6 | eqid 2798 | . . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
7 | 1, 2, 3, 4, 5, 6 | isphl 20317 | . . . 4 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑦 ∈ 𝑉 ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g‘𝐹) → 𝑦 = (0g‘𝑊)) ∧ ∀𝑥 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)))) |
8 | 7 | simp3bi 1144 | . . 3 ⊢ (𝑊 ∈ PreHil → ∀𝑦 ∈ 𝑉 ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g‘𝐹) → 𝑦 = (0g‘𝑊)) ∧ ∀𝑥 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦))) |
9 | simp1 1133 | . . . 4 ⊢ (((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g‘𝐹) → 𝑦 = (0g‘𝑊)) ∧ ∀𝑥 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) | |
10 | 9 | ralimi 3128 | . . 3 ⊢ (∀𝑦 ∈ 𝑉 ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g‘𝐹) → 𝑦 = (0g‘𝑊)) ∧ ∀𝑥 ∈ 𝑉 ((*𝑟‘𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)) → ∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
11 | 8, 10 | syl 17 | . 2 ⊢ (𝑊 ∈ PreHil → ∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
12 | oveq2 7143 | . . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴)) | |
13 | 12 | mpteq2dv 5126 | . . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))) |
14 | phllmhm.g | . . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)) | |
15 | 13, 14 | eqtr4di 2851 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = 𝐺) |
16 | 15 | eleq1d 2874 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))) |
17 | 16 | rspccva 3570 | . 2 ⊢ ((∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
18 | 11, 17 | sylan 583 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 *𝑟cstv 16559 Scalarcsca 16560 ·𝑖cip 16562 0gc0g 16705 *-Ringcsr 19608 LMHom clmhm 19784 LVecclvec 19867 ringLModcrglmod 19934 PreHilcphl 20313 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-iota 6283 df-fv 6332 df-ov 7138 df-phl 20315 |
This theorem is referenced by: ipcl 20322 ip0l 20325 ipdir 20328 ipass 20334 |
Copyright terms: Public domain | W3C validator |