llni - Mathbox for Norm Megill

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Theorem llni 37449

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 1190	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝐵)
2		breq1 5073	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝𝐶𝑋 ↔ 𝑃𝐶𝑋))
3	2	rspcev 3552	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
4	3	3ad2antl3 1185	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
5		simpl1 1189	. . 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 37447	. . 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 709	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 class class class wbr 5070 ‘cfv 6418 Basecbs 16840 ⋖ ccvr 37203 Atomscatm 37204 LLinesclln 37432

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-iota 6376 df-fun 6420 df-fv 6426 df-llines 37439

This theorem is referenced by: llnle 37459 atcvrlln 37461 lplncvrlvol 37557

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