Theorem ipcj 21784

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 2763	. . . . . 6 ⊢ (0g‘𝑊) = (0g‘𝑊)
5		ipcj.i	. . . . . 6 ⊢ ∗ = (*𝑟‘𝐹)
6		eqid 2763	. . . . . 6 ⊢ (0g‘𝐹) = (0g‘𝐹)
7	1, 2, 3, 4, 5, 6	isphl 21778	. . . . 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 1165	. . . 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 1156	. . . . 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 3102	. . . 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 18	. . 3 ⊢ (𝑊 ∈ PreHil → ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
12		fvoveq1 7433	. . . . 5 ⊢ (𝑥 = 𝐴 → ( ∗ ‘(𝑥 , 𝑦)) = ( ∗ ‘(𝐴 , 𝑦)))
13		oveq2 7418	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
14	12, 13	eqeq12d 2779	. . . 4 ⊢ (𝑥 = 𝐴 → (( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15		oveq2 7418	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
16	15	fveq2d 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → ( ∗ ‘(𝐴 , 𝑦)) = ( ∗ ‘(𝐴 , 𝐵)))
17		oveq1 7417	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
18	16, 17	eqeq12d 2779	. . . 4 ⊢ (𝑦 = 𝐵 → (( ∗ ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
19	14, 18	rspc2v 3592	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 ( ∗ ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
20	11, 19	syl5com 32	. 2 ⊢ (𝑊 ∈ PreHil → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
21	20	3impib 1134	1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ( ∗ ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ↦ cmpt 5192  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  *_𝑟cstv 17307  Scalarcsca 17308  ·_𝑖cip 17310  0_gc0g 17487  *-Ringcsr 20941   LMHom clmhm 21140  LVecclvec 21223  ringLModcrglmod 21293  PreHilcphl 21774

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 5269

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 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-iota 6492  df-fv 6544  df-ov 7413  df-phl 21776

This theorem is referenced by:  iporthcom  21785  ip0r  21787  ipdi  21790  ipassr  21796  phlssphl  21809  cphipcj  25358  tcphcphlem3  25392  ipcau2  25393  tcphcphlem1  25394