Theorem lineset 39858

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 3458	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		lineset.n	. . 3 ⊢ 𝑁 = (Lines‘𝐾)
3		fveq2 6828	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
4		lineset.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5	3, 4	eqtr4di 2786	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
6		fveq2 6828	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
7		lineset.l	. . . . . . . . . . . . 13 ⊢ ≤ = (le‘𝐾)
8	6, 7	eqtr4di 2786	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
9	8	breqd 5104	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞(join‘𝑘)𝑟)))
10		fveq2 6828	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
11		lineset.j	. . . . . . . . . . . . . 14 ⊢ ∨ = (join‘𝐾)
12	10, 11	eqtr4di 2786	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
13	12	oveqd 7369	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐾 → (𝑞(join‘𝑘)𝑟) = (𝑞 ∨ 𝑟))
14	13	breq2d 5105	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑝 ≤ (𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
15	9, 14	bitrd 279	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟) ↔ 𝑝 ≤ (𝑞 ∨ 𝑟)))
16	5, 15	rabeqbidv 3414	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
17	16	eqeq2d 2744	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑠 = {𝑝 ∈ (Atoms‘𝑘) ∣ 𝑝(le‘𝑘)(𝑞(join‘𝑘)𝑟)} ↔ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18	17	anbi2d 630	. . . . . . 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 3314	. . . . . 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 3314	. . . . 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 2799	. . . 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 39621	. . . 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 6842	. . . . 5 ⊢ 𝐴 ∈ V
24		df-sn 4576	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} = {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
25		snex 5376	. . . . . . 7 ⊢ {{𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
26	24, 25	eqeltrri 2830	. . . . . 6 ⊢ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}} ∈ V
27		simpr 484	. . . . . . 7 ⊢ ((𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) → 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})
28	27	ss2abi 4015	. . . . . 6 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ⊆ {𝑠 ∣ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}}
29	26, 28	ssexi 5262	. . . . 5 ⊢ {𝑠 ∣ (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
30	23, 23, 29	ab2rexex2 7918	. . . 4 ⊢ {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})} ∈ V
31	21, 22, 30	fvmpt 6935	. . 3 ⊢ (𝐾 ∈ V → (Lines‘𝐾) = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
32	2, 31	eqtrid 2780	. 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 395   = wceq 1541   ∈ wcel 2113  {cab 2711   ≠ wne 2929  ∃wrex 3057  {crab 3396  Vcvv 3437  {csn 4575   class class class wbr 5093  ‘cfv 6486  (class class class)co 7352  lecple 17170  joincjn 18219  Atomscatm 39383  Linesclines 39614

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 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7355  df-lines 39621

This theorem is referenced by:  isline  39859