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| Mirrors > Home > HSE Home > Th. List > chne0i | Structured version Visualization version GIF version | ||
| Description: A nonzero closed subspace has a nonzero vector. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| ch0le.1 | ⊢ 𝐴 ∈ Cℋ | 
| Ref | Expression | 
|---|---|
| chne0i | ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ch0le.1 | . . 3 ⊢ 𝐴 ∈ Cℋ | |
| 2 | 1 | chshii 31246 | . 2 ⊢ 𝐴 ∈ Sℋ | 
| 3 | 2 | shne0i 31467 | 1 ⊢ (𝐴 ≠ 0ℋ ↔ ∃𝑥 ∈ 𝐴 𝑥 ≠ 0ℎ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 0ℎc0v 30943 Cℋ cch 30948 0ℋc0h 30954 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-hilex 31018 ax-hv0cl 31022 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-ov 7434 df-sh 31226 df-ch 31240 df-ch0 31272 | 
| This theorem is referenced by: chne0 31513 hne0 31566 h1datomi 31600 riesz3i 32081 pjnmopi 32167 | 
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