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Mirrors > Home > HSE Home > Th. List > ax-his1 | Structured version Visualization version GIF version |
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. Note that ∗‘𝑥 is the complex conjugate cjval 14551 of 𝑥. In the literature, the inner product of 𝐴 and 𝐵 is usually written 〈𝐴, 𝐵〉, but our operation notation co 7170 allows us to use existing theorems about operations and also avoids a clash with the definition of an ordered pair df-op 4523. Physicists use 〈𝐵 ∣ 𝐴〉, called Dirac bra-ket notation, to represent this operation; see comments in df-bra 29785. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-his1 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | chba 28854 | . . . 4 class ℋ | |
3 | 1, 2 | wcel 2114 | . . 3 wff 𝐴 ∈ ℋ |
4 | cB | . . . 4 class 𝐵 | |
5 | 4, 2 | wcel 2114 | . . 3 wff 𝐵 ∈ ℋ |
6 | 3, 5 | wa 399 | . 2 wff (𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) |
7 | csp 28857 | . . . 4 class ·ih | |
8 | 1, 4, 7 | co 7170 | . . 3 class (𝐴 ·ih 𝐵) |
9 | 4, 1, 7 | co 7170 | . . . 4 class (𝐵 ·ih 𝐴) |
10 | ccj 14545 | . . . 4 class ∗ | |
11 | 9, 10 | cfv 6339 | . . 3 class (∗‘(𝐵 ·ih 𝐴)) |
12 | 8, 11 | wceq 1542 | . 2 wff (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)) |
13 | 6, 12 | wi 4 | 1 wff ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) |
Colors of variables: wff setvar class |
This axiom is referenced by: his5 29021 his7 29025 his2sub2 29028 hire 29029 hi02 29032 his1i 29035 abshicom 29036 hial2eq2 29042 orthcom 29043 adjsym 29768 cnvadj 29827 adj2 29869 |
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