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Theorem phllmhm 20195
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 2771 . . . . 5 (0g𝑊) = (0g𝑊)
5 eqid 2771 . . . . 5 (*𝑟𝐹) = (*𝑟𝐹)
6 eqid 2771 . . . . 5 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20191 . . . 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 1141 . . 3 (𝑊 ∈ PreHil → ∀𝑦𝑉 ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑦 , 𝑦) = (0g𝐹) → 𝑦 = (0g𝑊)) ∧ ∀𝑥𝑉 ((*𝑟𝐹)‘(𝑦 , 𝑥)) = (𝑥 , 𝑦)))
9 simp1 1130 . . . 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 3101 . . 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 6802 . . . . . 6 (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
1312mpteq2dv 4880 . . . . 5 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = (𝑥𝑉 ↦ (𝑥 , 𝐴)))
14 phllmhm.g . . . . 5 𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))
1513, 14syl6eqr 2823 . . . 4 (𝑦 = 𝐴 → (𝑥𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
1615eleq1d 2835 . . 3 (𝑦 = 𝐴 → ((𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
1716rspccva 3460 . 2 ((∀𝑦𝑉 (𝑥𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
1811, 17sylan 563 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  cmpt 4864  cfv 6032  (class class class)co 6794  Basecbs 16065  *𝑟cstv 16152  Scalarcsca 16153  ·𝑖cip 16155  0gc0g 16309  *-Ringcsr 19055   LMHom clmhm 19233  LVecclvec 19316  ringLModcrglmod 19385  PreHilcphl 20187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-iota 5995  df-fv 6040  df-ov 6797  df-phl 20189
This theorem is referenced by:  ipcl  20196  ip0l  20199  ipdir  20202  ipass  20208
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