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Theorem ipcj 20179
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 2758 . . . . . 6 (0g𝑊) = (0g𝑊)
5 ipcj.i . . . . . 6 = (*𝑟𝐹)
6 eqid 2758 . . . . . 6 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20173 . . . . 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 1142 . . . 4 (𝑊 ∈ PreHil → ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
9 simp3 1133 . . . . 5 (((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
109ralimi 3088 . . . 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 6818 . . . . . 6 (𝑥 = 𝐴 → (𝑥 , 𝑦) = (𝐴 , 𝑦))
1312fveq2d 6354 . . . . 5 (𝑥 = 𝐴 → ( ‘(𝑥 , 𝑦)) = ( ‘(𝐴 , 𝑦)))
14 oveq2 6819 . . . . 5 (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
1513, 14eqeq12d 2773 . . . 4 (𝑥 = 𝐴 → (( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
16 oveq2 6819 . . . . . 6 (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
1716fveq2d 6354 . . . . 5 (𝑦 = 𝐵 → ( ‘(𝐴 , 𝑦)) = ( ‘(𝐴 , 𝐵)))
18 oveq1 6818 . . . . 5 (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
1917, 18eqeq12d 2773 . . . 4 (𝑦 = 𝐵 → (( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2015, 19rspc2v 3459 . . 3 ((𝐴𝑉𝐵𝑉) → (∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2111, 20syl5com 31 . 2 (𝑊 ∈ PreHil → ((𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
22213impib 1109 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  cmpt 4879  cfv 6047  (class class class)co 6811  Basecbs 16057  *𝑟cstv 16143  Scalarcsca 16144  ·𝑖cip 16146  0gc0g 16300  *-Ringcsr 19044   LMHom clmhm 19219  LVecclvec 19302  ringLModcrglmod 19369  PreHilcphl 20169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-nul 4939
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-mpt 4880  df-iota 6010  df-fv 6055  df-ov 6814  df-phl 20171
This theorem is referenced by:  iporthcom  20180  ip0r  20182  ipdi  20185  ipassr  20191  cphipcj  23197  tchcphlem3  23230  ipcau2  23231  tchcphlem1  23232
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