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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 1191 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
2 | breq1 5151 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
3 | 2 | rspcev 3622 | . . 3 ⊢ ((𝑋 ∈ 𝑁 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
4 | 3 | 3ad2antl3 1186 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
5 | simpl1 1190 | . . 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 39513 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 class class class wbr 5148 ‘cfv 6563 Basecbs 17245 ⋖ ccvr 39244 LLinesclln 39474 LPlanesclpl 39475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lplanes 39482 |
This theorem is referenced by: lplnle 39523 llncvrlpln 39541 lplnexllnN 39547 |
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