Theorem lineset 35024

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 3212	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		lineset.n	. . 3 ⊢ 𝑁 = (Lines‘𝐾)
3		fveq2 6191	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		lineset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	syl6eqr 2674	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7		lineset.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
8	6, 7	syl6eqr 2674	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
9	8	breqd 4664	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞(join‘𝑘)𝑟)))
10		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11		lineset.j	. . . . . . . . . . . . . 14 ⊢ ∨ = (join‘𝐾)
12	10, 11	syl6eqr 2674	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
13	12	oveqd 6667	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 ∨ 𝑟))
14	13	breq2d 4665	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝 ≤ (𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
15	9, 14	bitrd 268	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	5, 15	rabeqbidv 3195	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
17	16	eqeq2d 2632	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18	17	anbi2d 740	. . . . . . 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 3153	. . . . . 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 3153	. . . . 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 2741	. . . 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 34787	. . . 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 6201	. . . . . 6 ⊢ (Atoms‘𝐾) ∈ V
24	4, 23	eqeltri 2697	. . . . 5 ⊢ 𝐴 ∈ V
25		df-sn 4178	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} = {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
26		snex 4908	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
27	25, 26	eqeltrri 2698	. . . . . 6 ⊢ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
28		simpr 477	. . . . . . 7 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
29	28	ss2abi 3674	. . . . . 6 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ⊆ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
30	27, 29	ssexi 4803	. . . . 5 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
31	24, 24, 30	ab2rexex2 7160	. . . 4 ⊢ {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
32	21, 22, 31	fvmpt 6282	. . 3 ⊢ (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
33	2, 32	syl5eq 2668	. 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 384   = wceq 1483   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∃wrex 2913  {crab 2916  Vcvv 3200  {csn 4177   class class class wbr 4653  ‘cfv 5888  (class class class)co 6650  lecple 15948  joincjn 16944  Atomscatm 34550  Linesclines 34780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-lines 34787

This theorem is referenced by:  isline  35025