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Mirrors > Home > HSE Home > Th. List > his1i | Structured version Visualization version GIF version |
Description: Conjugate law for inner product. Postulate (S1) of [Beran] p. 95. (Contributed by NM, 15-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
his1.1 | ⊢ 𝐴 ∈ ℋ |
his1.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
his1i | ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | his1.1 | . 2 ⊢ 𝐴 ∈ ℋ | |
2 | his1.2 | . 2 ⊢ 𝐵 ∈ ℋ | |
3 | ax-his1 28853 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴))) | |
4 | 1, 2, 3 | mp2an 690 | 1 ⊢ (𝐴 ·ih 𝐵) = (∗‘(𝐵 ·ih 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ∗ccj 14449 ℋchba 28690 ·ih csp 28693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-his1 28853 |
This theorem depends on definitions: df-bi 209 df-an 399 |
This theorem is referenced by: normlem2 28882 bcseqi 28891 bcsiALT 28950 pjadjii 29445 lnopunilem1 29781 lnophmlem2 29788 |
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