Theorem islinei 40399

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 1209	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑄 ∈ 𝐴)
2		simpl3 1210	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑅 ∈ 𝐴)
3		simpr 489	. . 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 3026	. . . . 5 ⊢ (𝑞 = 𝑄 → (𝑞 ≠ 𝑟 ↔ 𝑄 ≠ 𝑟))
5		oveq1 7415	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟))
6	5	breq2d 5122	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑟)))
7	6	rabbidv 3430	. . . . . 6 ⊢ (𝑞 = 𝑄 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)})
8	7	eqeq2d 2780	. . . . 5 ⊢ (𝑞 = 𝑄 → (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)}))
9	4, 8	anbi12d 643	. . . 4 ⊢ (𝑞 = 𝑄 → ((𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)}) ↔ (𝑄 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)})))
10		neeq2 3027	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑄 ≠ 𝑟 ↔ 𝑄 ≠ 𝑅))
11		oveq2 7416	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	11	breq2d 5122	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑝 ≤ (𝑄 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑅)))
13	12	rabbidv 3430	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})
14	13	eqeq2d 2780	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)} ↔ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)}))
15	10, 14	anbi12d 643	. . . 4 ⊢ (𝑟 = 𝑅 → ((𝑄 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)}) ↔ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})))
16	9, 15	rspc2ev 3603	. . 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 1396	. 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 1208	. . 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 40398	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
24	18, 23	syl 18	. 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 260	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑋 ∈ 𝑁)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∃wrex 3095  {crab 3423   class class class wbr 5110  ‘cfv 6534  (class class class)co 7408  lecple 17313  joincjn 18363  Atomscatm 39922  Linesclines 40153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-lines 40160

This theorem is referenced by:  linepmap  40434