Theorem islinei 34044

Description: Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)

Hypotheses

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

Assertion

Ref	Expression
islinei	⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑋 ∈ 𝑁)

Distinct variable groups:   𝐴,𝑝   𝐾,𝑝   𝑄,𝑝   𝑅,𝑝

Allowed substitution hints:   𝐷(𝑝)   ∨ (𝑝)   ≤ (𝑝)   𝑁(𝑝)   𝑋(𝑝)

Proof of Theorem islinei

Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1058	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑄 ∈ 𝐴)
2		simpl3 1059	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑅 ∈ 𝐴)
3		simpr 476	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)}))
4		neeq1 2844	. . . . 5 ⊢ (𝑞 = 𝑄 → (𝑞 ≠ 𝑟 ↔ 𝑄 ≠ 𝑟))
5		oveq1 6556	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟))
6	5	breq2d 4595	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑟)))
7	6	rabbidv 3164	. . . . . 6 ⊢ (𝑞 = 𝑄 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)})
8	7	eqeq2d 2620	. . . . 5 ⊢ (𝑞 = 𝑄 → (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)}))
9	4, 8	anbi12d 743	. . . 4 ⊢ (𝑞 = 𝑄 → ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑄 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)})))
10		neeq2 2845	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑄 ≠ 𝑟 ↔ 𝑄 ≠ 𝑅))
11		oveq2 6557	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	11	breq2d 4595	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑝 ≤ (𝑄 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑅)))
13	12	rabbidv 3164	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})
14	13	eqeq2d 2620	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)}))
15	10, 14	anbi12d 743	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑄 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)}) ↔ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})))
16	9, 15	rspc2ev 3295	. . 3 ⊢ ((𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
17	1, 2, 3, 16	syl3anc 1318	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}))
18		simpl1 1057	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝐾 ∈ 𝐷)
19		isline.l	. . . 4 ⊢ ≤ = (le‘𝐾)
20		isline.j	. . . 4 ⊢ ∨ = (join‘𝐾)
21		isline.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
22		isline.n	. . . 4 ⊢ 𝑁 = (Lines‘𝐾)
23	19, 20, 21, 22	isline 34043	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
24	18, 23	syl 17	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
25	17, 24	mpbird 246	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑋 ∈ 𝑁)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   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:  linepmap  34079