Theorem pointpsubN 39738

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 2729	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
2		pointpsub.p	. . . 4 ⊢ 𝑃 = (Points‘𝐾)
3	1, 2	ispointN 39729	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4		pointpsub.s	. . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 4	snatpsubN 39737	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
6	5	ex 412	. . . . 5 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7		eleq1a 2823	. . . . 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 3132	. . 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 3053  {csn 4585  ‘cfv 6499  Atomscatm 39249  AtLatcal 39250  PointscpointsN 39482  PSubSpcpsubsp 39483

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 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-proset 18235  df-poset 18254  df-plt 18269  df-lub 18285  df-glb 18286  df-join 18287  df-meet 18288  df-p0 18364  df-lat 18373  df-covers 39252  df-ats 39253  df-atl 39284  df-pointsN 39489  df-psubsp 39490

This theorem is referenced by: (None)