Theorem hlatmstcOLDN 38794

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 32146 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 38761	. 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 38715	. 2 ⊢ (((𝐾 ∈ OML ∧ 𝐾 ∈ CLat ∧ 𝐾 ∈ AtLat) ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)
7	1, 6	sylan 579	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑦 ∈ 𝐴 ∣ 𝑦 ≤ 𝑋}) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  {crab 3427   class class class wbr 5142  ‘cfv 6542  Basecbs 17165  lecple 17225  lubclub 18286  CLatccla 18475  OMLcoml 38571  Atomscatm 38659  AtLatcal 38660  HLchlt 38746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-proset 18272  df-poset 18290  df-plt 18307  df-lub 18323  df-glb 18324  df-join 18325  df-meet 18326  df-p0 18402  df-lat 18409  df-clat 18476  df-oposet 38572  df-ol 38574  df-oml 38575  df-covers 38662  df-ats 38663  df-atl 38694  df-cvlat 38718  df-hlat 38747

This theorem is referenced by: (None)