Theorem hlatmstcOLDN 39679

Description: An atomic, complete, orthomodular lattice is atomistic i.e. every element is the join of the atoms under it. See remark before Proposition 1 in [Kalmbach] p. 140; also remark in [BeltramettiCassinelli] p. 98. (hatomistici 32439 analog.) (Contributed by NM, 21-Oct-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hlatmstc.b	⊢ 𝐵 = (Base‘𝐾)
hlatmstc.l	⊢ ≤ = (le‘𝐾)
hlatmstc.u	⊢ 𝑈 = (lub‘𝐾)
hlatmstc.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
hlatmstcOLDN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦, ≤   𝑦,𝑋

Allowed substitution hints:   𝑈(𝑦)   𝐾(𝑦)

Proof of Theorem hlatmstcOLDN

Step	Hyp	Ref	Expression
1		hlomcmat 39647	. 2 ⊢ (𝐾 ∈ HL → (𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat))
2		hlatmstc.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
3		hlatmstc.l	. . 3 ⊢ ≤ = (le‘𝐾)
4		hlatmstc.u	. . 3 ⊢ 𝑈 = (lub‘𝐾)
5		hlatmstc.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6	2, 3, 4, 5	atlatmstc 39601	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
7	1, 6	sylan 580	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  {crab 3399   class class class wbr 5098  ‘cfv 6492  Basecbs 17138  lecple 17186  lubclub 18234  CLatccla 18423  OMLcoml 39457  Atomscatm 39545  AtLatcal 39546  HLchlt 39632

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 18219  df-poset 18238  df-plt 18253  df-lub 18269  df-glb 18270  df-join 18271  df-meet 18272  df-p0 18348  df-lat 18357  df-clat 18424  df-oposet 39458  df-ol 39460  df-oml 39461  df-covers 39548  df-ats 39549  df-atl 39580  df-cvlat 39604  df-hlat 39633

This theorem is referenced by: (None)