llni - Mathbox for Norm Megill

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Theorem llni 37519

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 1191	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝐵)
2		breq1 5079	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝𝐶𝑋 ↔ 𝑃𝐶𝑋))
3	2	rspcev 3561	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
4	3	3ad2antl3 1186	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
5		simpl1 1190	. . 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 37517	. . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))
11	5, 10	syl 17	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → (𝑋 ∈ 𝑁 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)))
12	1, 4, 11	mpbir2and 710	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∃wrex 3065 class class class wbr 5076 ‘cfv 6435 Basecbs 16910 ⋖ ccvr 37273 Atomscatm 37274 LLinesclln 37502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pr 5354

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-iota 6393 df-fun 6437 df-fv 6443 df-llines 37509

This theorem is referenced by: llnle 37529 atcvrlln 37531 lplncvrlvol 37627

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