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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 | โข ((๐ด โ โ โง ๐ด โ 0โ) โ 0 < (๐ด ยทih ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . . 4 class ๐ด | |
2 | chba 30172 | . . . 4 class โ | |
3 | 1, 2 | wcel 2107 | . . 3 wff ๐ด โ โ |
4 | c0v 30177 | . . . 4 class 0โ | |
5 | 1, 4 | wne 2941 | . . 3 wff ๐ด โ 0โ |
6 | 3, 5 | wa 397 | . 2 wff (๐ด โ โ โง ๐ด โ 0โ) |
7 | cc0 11110 | . . 3 class 0 | |
8 | csp 30175 | . . . 4 class ยทih | |
9 | 1, 1, 8 | co 7409 | . . 3 class (๐ด ยทih ๐ด) |
10 | clt 11248 | . . 3 class < | |
11 | 7, 9, 10 | wbr 5149 | . 2 wff 0 < (๐ด ยทih ๐ด) |
12 | 6, 11 | wi 4 | 1 wff ((๐ด โ โ โง ๐ด โ 0โ) โ 0 < (๐ด ยทih ๐ด)) |
Colors of variables: wff setvar class |
This axiom is referenced by: hiidge0 30351 his6 30352 normgt0 30380 eigrei 31087 eigposi 31089 |
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