Theorem phllmhm 21763

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 2763	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		eqid 2763	. . . . 5 ⊢ (*𝑟‘𝐹) = (*𝑟‘𝐹)
6		eqid 2763	. . . . 5 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21759	. . . 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 1165	. . 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 1154	. . . 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 3102	. . 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 18	. 2 ⊢ (𝑊 ∈ PreHil → ∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)))
12		oveq2 7420	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑥 , 𝑦) = (𝑥 , 𝐴))
13	12	mpteq2dv 5206	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴)))
14		phllmhm.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐴))
15	13, 14	eqtr4di 2816	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) = 𝐺)
16	15	eleq1d 2848	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ↔ 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹))))
17	16	rspccva 3581	. 2 ⊢ ((∀𝑦 ∈ 𝑉 (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝑦)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))
18	11, 17	sylan 591	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  *_𝑟cstv 17313  Scalarcsca 17314  ·_𝑖cip 17316  0_gc0g 17493  *-Ringcsr 20922   LMHom clmhm 21121  LVecclvec 21204  ringLModcrglmod 21274  PreHilcphl 21755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5270

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-iota 6494  df-fv 6546  df-ov 7415  df-phl 21757

This theorem is referenced by:  ipcl  21764  ip0l  21767  ipdir  21770  ipass  21776