Theorem phllmhm 21664

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	⊢ , = (·𝑖‘𝑊)
phllmhm.v	⊢ 𝑉 = (Base‘𝑊)
phllmhm.g	⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))

Assertion

Ref	Expression
phllmhm	⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 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.

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 2761	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2761	. . . . 5 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2761	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21660	. . . 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 1159	. . 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 1148	. . . 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 3098	. . 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 7400	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
13	12	mpteq2dv 5193	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)))
14		phllmhm.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))
15	13, 14	eqtr4di 2814	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
16	15	eleq1d 2846	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
17	16	rspccva 3580	. 2 ⊢ ((∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
18	11, 17	sylan 589	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075   ↦ cmpt 5180  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  *_𝑟cstv 17271  Scalarcsca 17272  ·_𝑖cip 17274  0_gc0g 17451  *-Ringcsr 20867   LMHom clmhm 21066  LVecclvec 21149  ringLModcrglmod 21219  PreHilcphl 21656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-nul 5255

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-iota 6473  df-fv 6525  df-ov 7395  df-phl 21658

This theorem is referenced by:  ipcl  21665  ip0l  21668  ipdir  21671  ipass  21677