Theorem lineset 36319

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 3427	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		lineset.n	. . 3 ⊢ 𝑁 = (Lines‘𝐾)
3		fveq2 6493	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		lineset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2826	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6493	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7		lineset.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
8	6, 7	syl6eqr 2826	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
9	8	breqd 4934	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞(join‘𝑘)𝑟)))
10		fveq2 6493	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11		lineset.j	. . . . . . . . . . . . . 14 ⊢ ∨ = (join‘𝐾)
12	10, 11	syl6eqr 2826	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
13	12	oveqd 6987	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 ∨ 𝑟))
14	13	breq2d 4935	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝 ≤ (𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
15	9, 14	bitrd 271	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	5, 15	rabeqbidv 3402	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
17	16	eqeq2d 2782	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18	17	anbi2d 619	. . . . . . 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 3336	. . . . . 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 3336	. . . . 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 2837	. . . 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 36082	. . . 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	4	fvexi 6507	. . . . 5 ⊢ 𝐴 ∈ V
24		df-sn 4436	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} = {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
25		snex 5182	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
26	24, 25	eqeltrri 2857	. . . . . 6 ⊢ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
27		simpr 477	. . . . . . 7 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
28	27	ss2abi 3927	. . . . . 6 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ⊆ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
29	26, 28	ssexi 5076	. . . . 5 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
30	23, 23, 29	ab2rexex2 7487	. . . 4 ⊢ {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
31	21, 22, 30	fvmpt 6589	. . 3 ⊢ (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
32	2, 31	syl5eq 2820	. 2 ⊢ (𝐾 ∈ V → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
33	1, 32	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050  {cab 2752   ≠ wne 2961  ∃wrex 3083  {crab 3086  Vcvv 3409  {csn 4435   class class class wbr 4923  ‘cfv 6182  (class class class)co 6970  lecple 16422  joincjn 17406  Atomscatm 35844  Linesclines 36075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pr 5180  ax-un 7273

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-lines 36082

This theorem is referenced by:  isline  36320