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Theorem phllmhm 21668
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 2735 . . . . 5 (0g𝑊) = (0g𝑊)
5 eqid 2735 . . . . 5 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2735 . . . . 5 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 21664 . . . 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 1146 . . 3 (𝑊 ∈ PreHil → ∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)))
9 simp1 1135 . . . 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 3081 . . 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 7439 . . . . . 6 (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
1312mpteq2dv 5250 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉 ↦ (𝑥 , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))
1513, 14eqtr4di 2793 . . . 4 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
1615eleq1d 2824 . . 3 (𝑦 = 𝐴 → ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
1716rspccva 3621 . 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 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  *𝑟cstv 17300  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  *-Ringcsr 20856   LMHom clmhm 21036  LVecclvec 21119  ringLModcrglmod 21189  PreHilcphl 21660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-phl 21662
This theorem is referenced by:  ipcl  21669  ip0l  21672  ipdir  21675  ipass  21681
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