Theorem ipcj 20839

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 2738	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		ipcj.i	. . . . . 6 ⊢ ∗ = (*𝑟‘𝐹)
6		eqid 2738	. . . . . 6 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 20833	. . . . 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 1146	. . . 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 1137	. . . . 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 3087	. . . 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 7298	. . . . 5 ⊢ (𝑥 = 𝐴 → ( ∗ ‘(𝑥 , 𝑦)) = ( ∗ ‘(𝐴 , 𝑦)))
13		oveq2 7283	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
14	12, 13	eqeq12d 2754	. . . 4 ⊢ (𝑥 = 𝐴 → (( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15		oveq2 7283	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
16	15	fveq2d 6778	. . . . 5 ⊢ (𝑦 = 𝐵 → ( ∗ ‘(𝐴 , 𝑦)) = ( ∗ ‘(𝐴 , 𝐵)))
17		oveq1 7282	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
18	16, 17	eqeq12d 2754	. . . 4 ⊢ (𝑦 = 𝐵 → (( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
19	14, 18	rspc2v 3570	. . 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 1115	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ↦ cmpt 5157  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  *_𝑟cstv 16964  Scalarcsca 16965  ·_𝑖cip 16967  0_gc0g 17150  *-Ringcsr 20104   LMHom clmhm 20281  LVecclvec 20364  ringLModcrglmod 20431  PreHilcphl 20829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-iota 6391  df-fv 6441  df-ov 7278  df-phl 20831

This theorem is referenced by:  iporthcom  20840  ip0r  20842  ipdi  20845  ipassr  20851  phlssphl  20864  cphipcj  24363  tcphcphlem3  24397  ipcau2  24398  tcphcphlem1  24399