Theorem phllmhm 20749

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 2738	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2738	. . . . 5 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2738	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 20745	. . . 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 1145	. . 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 1134	. . . 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 3086	. . 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 7263	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
13	12	mpteq2dv 5172	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)))
14		phllmhm.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))
15	13, 14	eqtr4di 2797	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
16	15	eleq1d 2823	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
17	16	rspccva 3551	. 2 ⊢ ((∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
18	11, 17	sylan 579	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  *_𝑟cstv 16890  Scalarcsca 16891  ·_𝑖cip 16893  0_gc0g 17067  *-Ringcsr 20019   LMHom clmhm 20196  LVecclvec 20279  ringLModcrglmod 20346  PreHilcphl 20741

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-iota 6376  df-fv 6426  df-ov 7258  df-phl 20743

This theorem is referenced by:  ipcl  20750  ip0l  20753  ipdir  20756  ipass  20762