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Theorem ipcj 20910
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 2737 . . . . . 6 (0g𝑊) = (0g𝑊)
5 ipcj.i . . . . . 6 = (*𝑟𝐹)
6 eqid 2737 . . . . . 6 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20904 . . . . 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 1146 . . . 4 (𝑊 ∈ PreHil → ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
9 simp3 1137 . . . . 5 (((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
109ralimi 3083 . . . 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 fvoveq1 7336 . . . . 5 (𝑥 = 𝐴 → ( ‘(𝑥 , 𝑦)) = ( ‘(𝐴 , 𝑦)))
13 oveq2 7321 . . . . 5 (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
1412, 13eqeq12d 2753 . . . 4 (𝑥 = 𝐴 → (( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15 oveq2 7321 . . . . . 6 (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
1615fveq2d 6813 . . . . 5 (𝑦 = 𝐵 → ( ‘(𝐴 , 𝑦)) = ( ‘(𝐴 , 𝐵)))
17 oveq1 7320 . . . . 5 (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
1816, 17eqeq12d 2753 . . . 4 (𝑦 = 𝐵 → (( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
1914, 18rspc2v 3579 . . 3 ((𝐴𝑉𝐵𝑉) → (∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2011, 19syl5com 31 . 2 (𝑊 ∈ PreHil → ((𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
21203impib 1115 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  cmpt 5168  cfv 6463  (class class class)co 7313  Basecbs 16979  *𝑟cstv 17031  Scalarcsca 17032  ·𝑖cip 17034  0gc0g 17217  *-Ringcsr 20175   LMHom clmhm 20352  LVecclvec 20435  ringLModcrglmod 20502  PreHilcphl 20900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2708  ax-nul 5243
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2942  df-ral 3063  df-rab 3405  df-v 3443  df-sbc 3726  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-br 5086  df-opab 5148  df-mpt 5169  df-iota 6415  df-fv 6471  df-ov 7316  df-phl 20902
This theorem is referenced by:  iporthcom  20911  ip0r  20913  ipdi  20916  ipassr  20922  phlssphl  20935  cphipcj  24434  tcphcphlem3  24468  ipcau2  24469  tcphcphlem1  24470
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