Theorem pointpsubN 39775

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.)

Hypotheses

Ref	Expression
pointpsub.p	⊢ 𝑃 = (Points‘𝐾)
pointpsub.s	⊢ 𝑆 = (PSubSp‘𝐾)

Assertion

Ref	Expression
pointpsubN	⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑆)

Proof of Theorem pointpsubN

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2736	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		pointpsub.p	. . . 4 ⊢ 𝑃 = (Points‘𝐾)
3	1, 2	ispointN 39766	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4		pointpsub.s	. . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 4	snatpsubN 39774	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
6	5	ex 412	. . . . 5 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7		eleq1a 2830	. . . . 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 3140	. . 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)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  {csn 4606  ‘cfv 6536  Atomscatm 39286  AtLatcal 39287  PointscpointsN 39519  PSubSpcpsubsp 39520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-proset 18311  df-poset 18330  df-plt 18345  df-lub 18361  df-glb 18362  df-join 18363  df-meet 18364  df-p0 18440  df-lat 18447  df-covers 39289  df-ats 39290  df-atl 39321  df-pointsN 39526  df-psubsp 39527

This theorem is referenced by: (None)