Theorem ipcj 20348

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 2825	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		ipcj.i	. . . . . 6 ⊢ ∗ = (*𝑟‘𝐹)
6		eqid 2825	. . . . . 6 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 20342	. . . . 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 1181	. . . 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 1172	. . . . 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 3161	. . . 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 6933	. . . . 5 ⊢ (𝑥 = 𝐴 → ( ∗ ‘(𝑥 , 𝑦)) = ( ∗ ‘(𝐴 , 𝑦)))
13		oveq2 6918	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
14	12, 13	eqeq12d 2840	. . . 4 ⊢ (𝑥 = 𝐴 → (( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15		oveq2 6918	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
16	15	fveq2d 6441	. . . . 5 ⊢ (𝑦 = 𝐵 → ( ∗ ‘(𝐴 , 𝑦)) = ( ∗ ‘(𝐴 , 𝐵)))
17		oveq1 6917	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
18	16, 17	eqeq12d 2840	. . . 4 ⊢ (𝑦 = 𝐵 → (( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
19	14, 18	rspc2v 3539	. . 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 1148	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  ∀wral 3117   ↦ cmpt 4954  ‘cfv 6127  (class class class)co 6910  Basecbs 16229  *_𝑟cstv 16314  Scalarcsca 16315  ·_𝑖cip 16317  0_gc0g 16460  *-Ringcsr 19207   LMHom clmhm 19385  LVecclvec 19468  ringLModcrglmod 19537  PreHilcphl 20338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-iota 6090  df-fv 6135  df-ov 6913  df-phl 20340

This theorem is referenced by:  iporthcom  20349  ip0r  20351  ipdi  20354  ipassr  20360  phlssphl  20373  cphipcj  23375  tcphcphlem3  23408  ipcau2  23409  tcphcphlem1  23410