Theorem lineset 34042

Description: The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)

Hypotheses

Ref	Expression
lineset.l	⊢ ≤ = (le‘𝐾)
lineset.j	⊢ ∨ = (join‘𝐾)
lineset.a	⊢ 𝐴 = (Atoms‘𝐾)
lineset.n	⊢ 𝑁 = (Lines‘𝐾)

Assertion

Ref	Expression
lineset	⊢ (𝐾 ∈ 𝐵 → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})

Distinct variable groups:   𝑞,𝑝,𝑟,𝑠,𝐴   𝐾,𝑝,𝑞,𝑟,𝑠   ∨ ,𝑠   ≤ ,𝑠

Allowed substitution hints:   𝐵(𝑠,𝑟,𝑞,𝑝)   ∨ (𝑟,𝑞,𝑝)   ≤ (𝑟,𝑞,𝑝)   𝑁(𝑠,𝑟,𝑞,𝑝)

Proof of Theorem lineset

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3185	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		lineset.n	. . 3 ⊢ 𝑁 = (Lines‘𝐾)
3		fveq2 6103	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		lineset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2662	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7		lineset.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
8	6, 7	syl6eqr 2662	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
9	8	breqd 4594	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞(join‘𝑘)𝑟)))
10		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11		lineset.j	. . . . . . . . . . . . . 14 ⊢ ∨ = (join‘𝐾)
12	10, 11	syl6eqr 2662	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
13	12	oveqd 6566	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 ∨ 𝑟))
14	13	breq2d 4595	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝 ≤ (𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
15	9, 14	bitrd 267	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	5, 15	rabeqbidv 3168	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
17	16	eqeq2d 2620	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18	17	anbi2d 736	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
19	5, 18	rexeqbidv 3130	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
20	5, 19	rexeqbidv 3130	. . . . 5 ⊢ (𝑘 = 𝐾 → (∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)}) ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
21	20	abbidv 2728	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})} = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
22		df-lines 33805	. . . 4 ⊢ Lines = (𝑘 ∈ V ↦ {𝑠 ∣ ∃𝑞 ∈ (Atoms‘𝑘)∃𝑟 ∈ (Atoms‘𝑘)(𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)})})
23		fvex 6113	. . . . . 6 ⊢ (Atoms‘𝐾) ∈ V
24	4, 23	eqeltri 2684	. . . . 5 ⊢ 𝐴 ∈ V
25		df-sn 4126	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} = {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
26		snex 4835	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
27	25, 26	eqeltrri 2685	. . . . . 6 ⊢ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
28		simpr 476	. . . . . . 7 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
29	28	ss2abi 3637	. . . . . 6 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ⊆ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
30	27, 29	ssexi 4731	. . . . 5 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
31	24, 24, 30	ab2rexex2 7051	. . . 4 ⊢ {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
32	21, 22, 31	fvmpt 6191	. . 3 ⊢ (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
33	2, 32	syl5eq 2656	. 2 ⊢ (𝐾 ∈ V → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
34	1, 33	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  lecple 15775  joincjn 16767  Atomscatm 33568  Linesclines 33798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-lines 33805

This theorem is referenced by:  isline  34043