llni - Mathbox for Norm Megill

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Theorem llni 35584

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 1250	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝐵)
2		breq1 4877	. . . 4 ⊢ (𝑝 = 𝑃 → (𝑝𝐶𝑋 ↔ 𝑃𝐶𝑋))
3	2	rspcev 3527	. . 3 ⊢ ((𝑃 ∈ 𝐴 ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
4	3	3ad2antl3 1244	. 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋)
5		simpl1 1248	. . 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 35582	. . 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 706	1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ 𝑃𝐶𝑋) → 𝑋 ∈ 𝑁)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ∃wrex 3119 class class class wbr 4874 ‘cfv 6124 Basecbs 16223 ⋖ ccvr 35338 Atomscatm 35339 LLinesclln 35567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pr 5128

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-sbc 3664 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-iota 6087 df-fun 6126 df-fv 6132 df-llines 35574

This theorem is referenced by: llnle 35594 atcvrlln 35596 lplncvrlvol 35692

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