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Axiom ax-his3 28485
Description: Associative law for inner product. Postulate (S3) of [Beran] p. 95. Warning: Mathematics textbooks usually use our version of the axiom. Physics textbooks, on the other hand, usually replace the left-hand side with (𝐵 ·ih (𝐴 · 𝐶)) (e.g., Equation 1.21b of [Hughes] p. 44; Definition (iii) of [ReedSimon] p. 36). See the comments in df-bra 29253 for why the physics definition is swapped. (Contributed by NM, 29-May-1999.) (New usage is discouraged.)
Assertion
Ref Expression
ax-his3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))

Detailed syntax breakdown of Axiom ax-his3
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 10250 . . . 4 class
31, 2wcel 2164 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
5 chba 28320 . . . 4 class
64, 5wcel 2164 . . 3 wff 𝐵 ∈ ℋ
7 cC . . . 4 class 𝐶
87, 5wcel 2164 . . 3 wff 𝐶 ∈ ℋ
93, 6, 8w3a 1111 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ)
10 csm 28322 . . . . 5 class ·
111, 4, 10co 6905 . . . 4 class (𝐴 · 𝐵)
12 csp 28323 . . . 4 class ·ih
1311, 7, 12co 6905 . . 3 class ((𝐴 · 𝐵) ·ih 𝐶)
144, 7, 12co 6905 . . . 4 class (𝐵 ·ih 𝐶)
15 cmul 10257 . . . 4 class ·
161, 14, 15co 6905 . . 3 class (𝐴 · (𝐵 ·ih 𝐶))
1713, 16wceq 1656 . 2 wff ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶))
189, 17wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 · 𝐵) ·ih 𝐶) = (𝐴 · (𝐵 ·ih 𝐶)))
Colors of variables: wff setvar class
This axiom is referenced by:  his5  28487  his35  28489  hiassdi  28492  his2sub  28493  hi01  28497  normlem0  28510  normlem9  28519  bcseqi  28521  polid2i  28558  ocsh  28686  h1de2i  28956  normcan  28979  eigrei  29237  eigorthi  29240  bramul  29349  lnopunilem1  29413  hmopm  29424  riesz3i  29465  cnlnadjlem2  29471  adjmul  29495  branmfn  29508  kbass2  29520  kbass5  29523  leopmuli  29536  leopnmid  29541
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