Theorem islinei 40514

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 1211	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑄 ∈ 𝐴)
2		simpl3 1212	. . 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 3020	. . . . 5 ⊢ (𝑞 = 𝑄 → (𝑞 ≠ 𝑟 ↔ 𝑄 ≠ 𝑟))
5		oveq1 7417	. . . . . . . 8 ⊢ (𝑞 = 𝑄 → (𝑞 ∨ 𝑟) = (𝑄 ∨ 𝑟))
6	5	breq2d 5121	. . . . . . 7 ⊢ (𝑞 = 𝑄 → (𝑝 ≤ (𝑞 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑟)))
7	6	rabbidv 3423	. . . . . 6 ⊢ (𝑞 = 𝑄 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)})
8	7	eqeq2d 2774	. . . . 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 3021	. . . . 5 ⊢ (𝑟 = 𝑅 → (𝑄 ≠ 𝑟 ↔ 𝑄 ≠ 𝑅))
11		oveq2 7418	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑄 ∨ 𝑟) = (𝑄 ∨ 𝑅))
12	11	breq2d 5121	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑝 ≤ (𝑄 ∨ 𝑟) ↔ 𝑝 ≤ (𝑄 ∨ 𝑅)))
13	12	rabbidv 3423	. . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑟)} = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})
14	13	eqeq2d 2774	. . . . 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 3594	. . 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 1398	. 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 1210	. . 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 40513	. . 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 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∃wrex 3089  {crab 3416   class class class wbr 5109  ‘cfv 6536  (class class class)co 7410  lecple 17312  joincjn 18362  Atomscatm 40037  Linesclines 40268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-lines 40275

This theorem is referenced by:  linepmap  40549