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Theorem phllmhm 20704
Description: The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
phlsrng.f 𝐹 = (Scalar‘𝑊)
phllmhm.h , = (·𝑖𝑊)
phllmhm.v 𝑉 = (Base‘𝑊)
phllmhm.g 𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))
Assertion
Ref Expression
phllmhm ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
Distinct variable groups:   𝑥,𝐴   𝑥, ,   𝑥,𝑉   𝑥,𝑊
Allowed substitution hints:   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem phllmhm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . 5 𝑉 = (Base‘𝑊)
2 phlsrng.f . . . . 5 𝐹 = (Scalar‘𝑊)
3 phllmhm.h . . . . 5 , = (·𝑖𝑊)
4 eqid 2818 . . . . 5 (0g𝑊) = (0g𝑊)
5 eqid 2818 . . . . 5 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2818 . . . . 5 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20700 . . . 4 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦))))
87simp3bi 1139 . . 3 (𝑊 ∈ PreHil → ∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)))
9 simp1 1128 . . . 4 (((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)) → (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)))
109ralimi 3157 . . 3 (∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)) → ∀𝑦𝑉 (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)))
118, 10syl 17 . 2 (𝑊 ∈ PreHil → ∀𝑦𝑉 (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)))
12 oveq2 7153 . . . . . 6 (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
1312mpteq2dv 5153 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉 ↦ (𝑥 , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))
1513, 14syl6eqr 2871 . . . 4 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
1615eleq1d 2894 . . 3 (𝑦 = 𝐴 → ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
1716rspccva 3619 . 2 ((∀𝑦𝑉 (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
1811, 17sylan 580 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  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:  ipcl  20705  ip0l  20708  ipdir  20711  ipass  20717
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