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Theorem ipcj 20181
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 2760 . . . . . 6 (0g𝑊) = (0g𝑊)
5 ipcj.i . . . . . 6 = (*𝑟𝐹)
6 eqid 2760 . . . . . 6 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20175 . . . . 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 3090 . . . 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 6820 . . . . . 6 (𝑥 = 𝐴 → (𝑥 , 𝑦) = (𝐴 , 𝑦))
1312fveq2d 6356 . . . . 5 (𝑥 = 𝐴 → ( ‘(𝑥 , 𝑦)) = ( ‘(𝐴 , 𝑦)))
14 oveq2 6821 . . . . 5 (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
1513, 14eqeq12d 2775 . . . 4 (𝑥 = 𝐴 → (( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
16 oveq2 6821 . . . . . 6 (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
1716fveq2d 6356 . . . . 5 (𝑦 = 𝐵 → ( ‘(𝐴 , 𝑦)) = ( ‘(𝐴 , 𝐵)))
18 oveq1 6820 . . . . 5 (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
1917, 18eqeq12d 2775 . . . 4 (𝑦 = 𝐵 → (( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2015, 19rspc2v 3461 . . 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 1632  wcel 2139  wral 3050  cmpt 4881  cfv 6049  (class class class)co 6813  Basecbs 16059  *𝑟cstv 16145  Scalarcsca 16146  ·𝑖cip 16148  0gc0g 16302  *-Ringcsr 19046   LMHom clmhm 19221  LVecclvec 19304  ringLModcrglmod 19371  PreHilcphl 20171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-iota 6012  df-fv 6057  df-ov 6816  df-phl 20173
This theorem is referenced by:  iporthcom  20182  ip0r  20184  ipdi  20187  ipassr  20193  cphipcj  23199  tchcphlem3  23232  ipcau2  23233  tchcphlem1  23234
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