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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 29330 | . . . . 5 ⊢ (𝐴 ∈ ℋ → 𝐴 ∈ (⊥‘(⊥‘{𝐴}))) | |
2 | eleq2 2903 | . . . . 5 ⊢ ((⊥‘(⊥‘{𝐴})) = 0ℋ → (𝐴 ∈ (⊥‘(⊥‘{𝐴})) ↔ 𝐴 ∈ 0ℋ)) | |
3 | 1, 2 | syl5ibcom 247 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((⊥‘(⊥‘{𝐴})) = 0ℋ → 𝐴 ∈ 0ℋ)) |
4 | elch0 29033 | . . . 4 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
5 | 3, 4 | syl6ib 253 | . . 3 ⊢ (𝐴 ∈ ℋ → ((⊥‘(⊥‘{𝐴})) = 0ℋ → 𝐴 = 0ℎ)) |
6 | 5 | necon3d 3039 | . 2 ⊢ (𝐴 ∈ ℋ → (𝐴 ≠ 0ℎ → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ)) |
7 | 6 | imp 409 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ) → (⊥‘(⊥‘{𝐴})) ≠ 0ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 {csn 4569 ‘cfv 6357 ℋchba 28698 0ℎc0v 28703 ⊥cort 28709 0ℋc0h 28714 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-hilex 28778 ax-hfvadd 28779 ax-hv0cl 28782 ax-hfvmul 28784 ax-hvmul0 28789 ax-hfi 28858 ax-his1 28861 ax-his2 28862 ax-his3 28863 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-2 11703 df-cj 14460 df-re 14461 df-im 14462 df-sh 28986 df-oc 29031 df-ch0 29032 |
This theorem is referenced by: h1da 30128 atom1d 30132 |
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