Theorem pointpsubN 40021

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 40012	. . 3 ⊢ (𝐾 ∈ AtLat → (𝑋 ∈ 𝑃 ↔ ∃𝑞 ∈ (Atoms‘𝐾)𝑋 = {𝑞}))
4		pointpsub.s	. . . . . . 7 ⊢ 𝑆 = (PSubSp‘𝐾)
5	1, 4	snatpsubN 40020	. . . . . 6 ⊢ ((𝐾 ∈ AtLat ∧ 𝑞 ∈ (Atoms‘𝐾)) → {𝑞} ∈ 𝑆)
6	5	ex 412	. . . . 5 ⊢ (𝐾 ∈ AtLat → (𝑞 ∈ (Atoms‘𝐾) → {𝑞} ∈ 𝑆))
7		eleq1a 2831	. . . . 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 3135	. . 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 1541   ∈ wcel 2113  ∃wrex 3060  {csn 4580  ‘cfv 6492  Atomscatm 39533  AtLatcal 39534  PointscpointsN 39765  PSubSpcpsubsp 39766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-proset 18217  df-poset 18236  df-plt 18251  df-lub 18267  df-glb 18268  df-join 18269  df-meet 18270  df-p0 18346  df-lat 18355  df-covers 39536  df-ats 39537  df-atl 39568  df-pointsN 39772  df-psubsp 39773

This theorem is referenced by: (None)