Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > ax-his4 | Structured version Visualization version GIF version |
Description: Identity law for inner product. Postulate (S4) of [Beran] p. 95. (Contributed by NM, 29-May-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax-his4 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class 𝐴 | |
2 | chil 27776 | . . . 4 class ℋ | |
3 | 1, 2 | wcel 1990 | . . 3 wff 𝐴 ∈ ℋ |
4 | c0v 27781 | . . . 4 class 0ℎ | |
5 | 1, 4 | wne 2794 | . . 3 wff 𝐴 ≠ 0ℎ |
6 | 3, 5 | wa 384 | . 2 wff (𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) |
7 | cc0 9936 | . . 3 class 0 | |
8 | csp 27779 | . . . 4 class ·ih | |
9 | 1, 1, 8 | co 6650 | . . 3 class (𝐴 ·ih 𝐴) |
10 | clt 10074 | . . 3 class < | |
11 | 7, 9, 10 | wbr 4653 | . 2 wff 0 < (𝐴 ·ih 𝐴) |
12 | 6, 11 | wi 4 | 1 wff ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → 0 < (𝐴 ·ih 𝐴)) |
Colors of variables: wff setvar class |
This axiom is referenced by: hiidge0 27955 his6 27956 normgt0 27984 eigrei 28693 eigposi 28695 |
Copyright terms: Public domain | W3C validator |