llni - Mathbox for Norm Megill

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Theorem llni 40381

Description: Condition implying a lattice line. (Contributed by NM, 17-Jun-2012.)

**Hypotheses**
Ref	Expression
llnset.b	⊢ 𝐵 = (Base‘𝐾)
llnset.c	⊢ 𝐶 = ( ⋖ ‘𝐾)
llnset.a	⊢ 𝐴 = (Atoms‘𝐾)
llnset.n	⊢ 𝑁 = (LLines‘𝐾)

**Assertion**
Ref	Expression
llni	⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)

Proof of Theorem llni Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl2 1211	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝐵)
2		breq1 5106	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝𝐶𝑋 ↔ 𝑃𝐶𝑋))
3	2	rspcev 3576	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
4	3	3ad2antl3 1206	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
5		simpl1 1210	. . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝐾 ∈ 𝐷)
6		llnset.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
7		llnset.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
8		llnset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
9		llnset.n	. . . 4 ⊢ 𝑁 = (LLines‘𝐾)
10	6, 7, 8, 9	islln 40379	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))
11	5, 10	syl 18	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))
12	1, 4, 11	mpbir2and 726	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∃wrex 3086 class class class wbr 5103 ‘cfv 6533 Basecbs 17301 ⋖ ccvr 40135 Atomscatm 40136 LLinesclln 40364

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-llines 40371

This theorem is used by: llnle 40391 atcvrlln 40393 lplncvrlvol 40489

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