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| Mirrors > Home > MPE Home > Th. List > Mathboxes > pointpsubN | Structured version Visualization version GIF version | ||
| Description: A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| pointpsub.p | ⊢ 𝑃 = (Points‘𝐾) | 
| pointpsub.s | ⊢ 𝑆 = (PSubSp‘𝐾) | 
| Ref | Expression | 
|---|---|
| pointpsubN | ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
| 2 | pointpsub.p | . . . 4 ⊢ 𝑃 = (Points‘𝐾) | |
| 3 | 1, 2 | ispointN 39745 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞})) | 
| 4 | pointpsub.s | . . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾) | |
| 5 | 1, 4 | snatpsubN 39753 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆) | 
| 6 | 5 | ex 412 | . . . . 5 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆)) | 
| 7 | eleq1a 2835 | . . . . 5 ⊢ ({𝑞} ∈ 𝑆 → (𝑋 = {𝑞} → 𝑋 ∈ 𝑆)) | |
| 8 | 6, 7 | syl6 35 | . . . 4 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → (𝑋 = {𝑞} → 𝑋 ∈ 𝑆))) | 
| 9 | 8 | rexlimdv 3152 | . . 3 ⊢ (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋 ∈ 𝑆)) | 
| 10 | 3, 9 | sylbid 240 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑆)) | 
| 11 | 10 | imp 406 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 {csn 4625 ‘cfv 6560 Atomscatm 39265 AtLatcal 39266 PointscpointsN 39498 PSubSpcpsubsp 39499 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-proset 18341 df-poset 18360 df-plt 18376 df-lub 18392 df-glb 18393 df-join 18394 df-meet 18395 df-p0 18471 df-lat 18478 df-covers 39268 df-ats 39269 df-atl 39300 df-pointsN 39505 df-psubsp 39506 | 
| This theorem is referenced by: (None) | 
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