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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 2734 | . . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
2 | pointpsub.p | . . . 4 ⊢ 𝑃 = (Points‘𝐾) | |
3 | 1, 2 | ispointN 37450 | . . 3 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞})) |
4 | pointpsub.s | . . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾) | |
5 | 1, 4 | snatpsubN 37458 | . . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆) |
6 | 5 | ex 416 | . . . . 5 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆)) |
7 | eleq1a 2829 | . . . . 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 3195 | . . 3 ⊢ (𝐾 ∈ AtLat → (∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞} → 𝑋 ∈ 𝑆)) |
10 | 3, 9 | sylbid 243 | . 2 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑆)) |
11 | 10 | imp 410 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3055 {csn 4531 ‘cfv 6369 Atomscatm 36971 AtLatcal 36972 PointscpointsN 37203 PSubSpcpsubsp 37204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-proset 17774 df-poset 17792 df-plt 17808 df-lub 17824 df-glb 17825 df-join 17826 df-meet 17827 df-p0 17903 df-lat 17910 df-covers 36974 df-ats 36975 df-atl 37006 df-pointsN 37210 df-psubsp 37211 |
This theorem is referenced by: (None) |
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