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Mirrors > Home > HSE Home > Th. List > h1dn0 | Structured version Visualization version GIF version |
Description: A nonzero vector generates a (nonzero) 1-dimensional subspace. (Contributed by NM, 22-Jul-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h1dn0 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h1did 29255 | . . . . 5 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴}))) | |
2 | eleq2 2898 | . . . . 5 ⊢ ((⊥‘(⊥‘{𝐴})) = 0ℋ → (𝐴 ∈ (⊥‘(⊥‘{𝐴})) ↔ 𝐴 ∈ 0ℋ)) | |
3 | 1, 2 | syl5ibcom 246 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((⊥‘(⊥‘{𝐴})) = 0ℋ → 𝐴 ∈ 0ℋ)) |
4 | elch0 28958 | . . . 4 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
5 | 3, 4 | syl6ib 252 | . . 3 ⊢ (𝐴 ∈ ℋ → ((⊥‘(⊥‘{𝐴})) = 0ℋ → 𝐴 = 0ℎ)) |
6 | 5 | necon3d 3034 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ)) |
7 | 6 | imp 407 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 {csn 4557 ‘cfv 6348 ℋchba 28623 0ℎc0v 28628 ⊥cort 28634 0ℋc0h 28639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-hilex 28703 ax-hfvadd 28704 ax-hv0cl 28707 ax-hfvmul 28709 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-cj 14446 df-re 14447 df-im 14448 df-sh 28911 df-oc 28956 df-ch0 28957 |
This theorem is referenced by: h1da 30053 atom1d 30057 |
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