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Mathbox for Norm Megill |
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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 1189 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
2 | breq1 5152 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
3 | 2 | rspcev 3606 | . . 3 ⊢ ((𝑋 ∈ 𝑁 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
4 | 3 | 3ad2antl3 1184 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
5 | simpl1 1188 | . . 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 39133 | . . 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 711 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 class class class wbr 5149 ‘cfv 6549 Basecbs 17183 ⋖ ccvr 38864 LLinesclln 39094 LPlanesclpl 39095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-lplanes 39102 |
This theorem is referenced by: lplnle 39143 llncvrlpln 39161 lplnexllnN 39167 |
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