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Theorem ipcj 20754
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 2821 . . . . . 6 (0g𝑊) = (0g𝑊)
5 ipcj.i . . . . . 6 = (*𝑟𝐹)
6 eqid 2821 . . . . . 6 (0g𝐹) = (0g𝐹)
71, 2, 3, 4, 5, 6isphl 20748 . . . . 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 1144 . . . 4 (𝑊 ∈ PreHil → ∀𝑥𝑉 ((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)))
9 simp3 1135 . . . . 5 (((𝑦𝑉 ↦ (𝑦 , 𝑥)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) ∧ ((𝑥 , 𝑥) = (0g𝐹) → 𝑥 = (0g𝑊)) ∧ ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥)) → ∀𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥))
109ralimi 3148 . . . 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 7153 . . . . 5 (𝑥 = 𝐴 → ( ‘(𝑥 , 𝑦)) = ( ‘(𝐴 , 𝑦)))
13 oveq2 7138 . . . . 5 (𝑥 = 𝐴 → (𝑦 , 𝑥) = (𝑦 , 𝐴))
1412, 13eqeq12d 2837 . . . 4 (𝑥 = 𝐴 → (( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) ↔ ( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴)))
15 oveq2 7138 . . . . . 6 (𝑦 = 𝐵 → (𝐴 , 𝑦) = (𝐴 , 𝐵))
1615fveq2d 6647 . . . . 5 (𝑦 = 𝐵 → ( ‘(𝐴 , 𝑦)) = ( ‘(𝐴 , 𝐵)))
17 oveq1 7137 . . . . 5 (𝑦 = 𝐵 → (𝑦 , 𝐴) = (𝐵 , 𝐴))
1816, 17eqeq12d 2837 . . . 4 (𝑦 = 𝐵 → (( ‘(𝐴 , 𝑦)) = (𝑦 , 𝐴) ↔ ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
1914, 18rspc2v 3610 . . 3 ((𝐴𝑉𝐵𝑉) → (∀𝑥𝑉𝑦𝑉 ( ‘(𝑥 , 𝑦)) = (𝑦 , 𝑥) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
2011, 19syl5com 31 . 2 (𝑊 ∈ PreHil → ((𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)))
21203impib 1113 1 ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  cmpt 5119  cfv 6328  (class class class)co 7130  Basecbs 16462  *𝑟cstv 16546  Scalarcsca 16547  ·𝑖cip 16549  0gc0g 16692  *-Ringcsr 19591   LMHom clmhm 19767  LVecclvec 19850  ringLModcrglmod 19917  PreHilcphl 20744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-nul 5183
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-iota 6287  df-fv 6336  df-ov 7133  df-phl 20746
This theorem is referenced by:  iporthcom  20755  ip0r  20757  ipdi  20760  ipassr  20766  phlssphl  20779  cphipcj  23783  tcphcphlem3  23816  ipcau2  23817  tcphcphlem1  23818
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