Theorem ipcj 21587

Description: Conjugate of an inner product in a pre-Hilbert space. Equation I1 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)

Hypotheses

Ref	Expression
phlsrng.f	⊢ 𝐹 = (Scalar‘𝑊)
phllmhm.h	⊢ , = (·𝑖‘𝑊)
phllmhm.v	⊢ 𝑉 = (Base‘𝑊)
ipcj.i	⊢ ∗ = (*𝑟‘𝐹)

Assertion

Ref	Expression
ipcj	⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Proof of Theorem ipcj

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		phllmhm.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑊)
2		phlsrng.f	. . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊)
3		phllmhm.h	. . . . . 6 ⊢ , = (·𝑖‘𝑊)
4		eqid 2734	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		ipcj.i	. . . . . 6 ⊢ ∗ = (*𝑟‘𝐹)
6		eqid 2734	. . . . . 6 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21581	. . . . 5 ⊢ (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
8	7	simp3bi 1147	. . . 4 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
9		simp3 1138	. . . . 5 ⊢ (((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
10	9	ralimi 3071	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 ((𝑦 ∈ 𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g‘𝐹) → 𝑥 = (0g‘𝑊)) ∧ ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
11	8, 10	syl 17	. . 3 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
12		fvoveq1 7379	. . . . 5 ⊢ (𝑥 = 𝐴 → ( ∗ ‘(𝑥 , 𝑦)) = ( ∗ ‘(𝐴 , 𝑦)))
13		oveq2 7364	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
14	12, 13	eqeq12d 2750	. . . 4 ⊢ (𝑥 = 𝐴 → (( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15		oveq2 7364	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
16	15	fveq2d 6836	. . . . 5 ⊢ (𝑦 = 𝐵 → ( ∗ ‘(𝐴 , 𝑦)) = ( ∗ ‘(𝐴 , 𝐵)))
17		oveq1 7363	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
18	16, 17	eqeq12d 2750	. . . 4 ⊢ (𝑦 = 𝐵 → (( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
19	14, 18	rspc2v 3585	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
20	11, 19	syl5com 31	. 2 ⊢ (𝑊 ∈ PreHil → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
21	20	3impib 1116	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ↦ cmpt 5177  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  *_𝑟cstv 17177  Scalarcsca 17178  ·_𝑖cip 17180  0_gc0g 17357  *-Ringcsr 20769   LMHom clmhm 20969  LVecclvec 21052  ringLModcrglmod 21122  PreHilcphl 21577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-nul 5249

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-iota 6446  df-fv 6498  df-ov 7359  df-phl 21579

This theorem is referenced by:  iporthcom  21588  ip0r  21590  ipdi  21593  ipassr  21599  phlssphl  21612  cphipcj  25153  tcphcphlem3  25187  ipcau2  25188  tcphcphlem1  25189