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Theorem phllmhm 20751
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 2821 . . . . 5 (0g𝑊) = (0g𝑊)
5 eqid 2821 . . . . 5 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2821 . . . . 5 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20747 . . . 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 1144 . . 3 (𝑊 ∈ PreHil → ∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)))
9 simp1 1133 . . . 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 3148 . . 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 7138 . . . . . 6 (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
1312mpteq2dv 5135 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉 ↦ (𝑥 , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))
1513, 14syl6eqr 2874 . . . 4 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
1615eleq1d 2896 . . 3 (𝑦 = 𝐴 → ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
1716rspccva 3599 . 2 ((∀𝑦𝑉 (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
1811, 17sylan 583 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cmpt 5119  cfv 6328  (class class class)co 7130  Basecbs 16461  *𝑟cstv 16545  Scalarcsca 16546  ·𝑖cip 16548  0gc0g 16691  *-Ringcsr 19590   LMHom clmhm 19766  LVecclvec 19849  ringLModcrglmod 19916  PreHilcphl 20743
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-iota 6287  df-fv 6336  df-ov 7133  df-phl 20745
This theorem is referenced by:  ipcl  20752  ip0l  20755  ipdir  20758  ipass  20764
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