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Theorem ipcj 19798
 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 , = (·𝑖𝑊)
phllmhm.v 𝑉 = (Base‘𝑊)
ipcj.i = (*𝑟𝐹)
Assertion
Ref Expression
ipcj ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Proof of Theorem ipcj
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 phllmhm.v . . . . . 6 𝑉 = (Base‘𝑊)
2 phlsrng.f . . . . . 6 𝐹 = (Scalar‘𝑊)
3 phllmhm.h . . . . . 6 , = (·𝑖𝑊)
4 eqid 2610 . . . . . 6 (0g𝑊) = (0g𝑊)
5 ipcj.i . . . . . 6 = (*𝑟𝐹)
6 eqid 2610 . . . . . 6 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 19792 . . . . 5 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ 𝐹 ∈ *-Ring ∧ ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))))
87simp3bi 1071 . . . 4 (𝑊 ∈ PreHil → ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
9 simp3 1056 . . . . 5 (((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
109ralimi 2936 . . . 4 (∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
118, 10syl 17 . . 3 (𝑊 ∈ PreHil → ∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
12 oveq1 6556 . . . . . 6 (𝑥 = 𝐴 → (𝑥 , 𝑦) = (𝐴 , 𝑦))
1312fveq2d 6107 . . . . 5 (𝑥 = 𝐴 → ( ‘(𝑥 , 𝑦)) = ( ‘(𝐴 , 𝑦)))
14 oveq2 6557 . . . . 5 (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
1513, 14eqeq12d 2625 . . . 4 (𝑥 = 𝐴 → (( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
16 oveq2 6557 . . . . . 6 (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
1716fveq2d 6107 . . . . 5 (𝑦 = 𝐵 → ( ‘(𝐴 , 𝑦)) = ( ‘(𝐴 , 𝐵)))
18 oveq1 6556 . . . . 5 (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
1917, 18eqeq12d 2625 . . . 4 (𝑦 = 𝐵 → (( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2015, 19rspc2v 3293 . . 3 ((𝐴𝑉𝐵𝑉) → (∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2111, 20syl5com 31 . 2 (𝑊 ∈ PreHil → ((𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
22213impib 1254 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  *𝑟cstv 15770  Scalarcsca 15771  ·𝑖cip 15773  0gc0g 15923  *-Ringcsr 18667   LMHom clmhm 18840  LVecclvec 18923  ringLModcrglmod 18990  PreHilcphl 19788 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-phl 19790 This theorem is referenced by:  iporthcom  19799  ip0r  19801  ipdi  19804  ipassr  19810  cphipcj  22807  tchcphlem3  22840  ipcau2  22841  tchcphlem1  22842
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