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Mirrors > Home > HSE Home > Th. List > ocnel | Structured version Visualization version GIF version |
Description: A nonzero vector in the complement of a subspace does not belong to the subspace. (Contributed by NM, 10-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ocnel | ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0ℎ) → ¬ 𝐴 ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3897 | . . . . . . 7 ⊢ (𝐴 ∈ (𝐻 ∩ (⊥‘𝐻)) ↔ (𝐴 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻))) | |
2 | ocin 29079 | . . . . . . . . 9 ⊢ (𝐻 ∈ Sℋ → (𝐻 ∩ (⊥‘𝐻)) = 0ℋ) | |
3 | 2 | eleq2d 2875 | . . . . . . . 8 ⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (𝐻 ∩ (⊥‘𝐻)) ↔ 𝐴 ∈ 0ℋ)) |
4 | 3 | biimpd 232 | . . . . . . 7 ⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (𝐻 ∩ (⊥‘𝐻)) → 𝐴 ∈ 0ℋ)) |
5 | 1, 4 | syl5bir 246 | . . . . . 6 ⊢ (𝐻 ∈ Sℋ → ((𝐴 ∈ 𝐻 ∧ 𝐴 ∈ (⊥‘𝐻)) → 𝐴 ∈ 0ℋ)) |
6 | 5 | expcomd 420 | . . . . 5 ⊢ (𝐻 ∈ Sℋ → (𝐴 ∈ (⊥‘𝐻) → (𝐴 ∈ 𝐻 → 𝐴 ∈ 0ℋ))) |
7 | 6 | imp 410 | . . . 4 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻)) → (𝐴 ∈ 𝐻 → 𝐴 ∈ 0ℋ)) |
8 | elch0 29037 | . . . 4 ⊢ (𝐴 ∈ 0ℋ ↔ 𝐴 = 0ℎ) | |
9 | 7, 8 | syl6ib 254 | . . 3 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻)) → (𝐴 ∈ 𝐻 → 𝐴 = 0ℎ)) |
10 | 9 | necon3ad 3000 | . 2 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻)) → (𝐴 ≠ 0ℎ → ¬ 𝐴 ∈ 𝐻)) |
11 | 10 | 3impia 1114 | 1 ⊢ ((𝐻 ∈ Sℋ ∧ 𝐴 ∈ (⊥‘𝐻) ∧ 𝐴 ≠ 0ℎ) → ¬ 𝐴 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∩ cin 3880 ‘cfv 6324 0ℎc0v 28707 Sℋ csh 28711 ⊥cort 28713 0ℋc0h 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hilex 28782 ax-hfvadd 28783 ax-hv0cl 28786 ax-hfvmul 28788 ax-hvmul0 28793 ax-hfi 28862 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sh 28990 df-oc 29035 df-ch0 29036 |
This theorem is referenced by: (None) |
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