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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lplni | Structured version Visualization version GIF version | ||
| Description: Condition implying a lattice plane. (Contributed by NM, 20-Jun-2012.) | 
| Ref | Expression | 
|---|---|
| lplnset.b | ⊢ 𝐵 = (Base‘𝐾) | 
| lplnset.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) | 
| lplnset.n | ⊢ 𝑁 = (LLines‘𝐾) | 
| lplnset.p | ⊢ 𝑃 = (LPlanes‘𝐾) | 
| Ref | Expression | 
|---|---|
| lplni | ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑃) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | simpl2 1193 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
| 2 | breq1 5146 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
| 3 | 2 | rspcev 3622 | . . 3 ⊢ ((𝑋 ∈ 𝑁 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) | 
| 4 | 3 | 3ad2antl3 1188 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) | 
| 5 | simpl1 1192 | . . 3 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ 𝐷) | |
| 6 | lplnset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
| 7 | lplnset.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
| 8 | lplnset.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
| 9 | lplnset.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
| 10 | 6, 7, 8, 9 | islpln 39532 | . . 3 ⊢ (𝐾 ∈ 𝐷 → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌))) | 
| 11 | 5, 10 | syl 17 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → (𝑌 ∈ 𝑃 ↔ (𝑌 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌))) | 
| 12 | 1, 4, 11 | mpbir2and 713 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑃) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 ⋖ ccvr 39263 LLinesclln 39493 LPlanesclpl 39494 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lplanes 39501 | 
| This theorem is referenced by: lplnle 39542 llncvrlpln 39560 lplnexllnN 39566 | 
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