Theorem phllmhm 21612

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 2736	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2736	. . . . 5 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2736	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21608	. . . 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 1148	. . 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 1137	. . . 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 3074	. . 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 7375	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
13	12	mpteq2dv 5179	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)))
14		phllmhm.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))
15	13, 14	eqtr4di 2789	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
16	15	eleq1d 2821	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
17	16	rspccva 3563	. 2 ⊢ ((∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
18	11, 17	sylan 581	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ↦ cmpt 5166  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  *_𝑟cstv 17222  Scalarcsca 17223  ·_𝑖cip 17225  0_gc0g 17402  *-Ringcsr 20815   LMHom clmhm 21014  LVecclvec 21097  ringLModcrglmod 21167  PreHilcphl 21604

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-nul 5241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-iota 6454  df-fv 6506  df-ov 7370  df-phl 21606

This theorem is referenced by:  ipcl  21613  ip0l  21616  ipdir  21619  ipass  21625