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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 1185 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵) | |
2 | breq1 4969 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐶𝑌 ↔ 𝑋𝐶𝑌)) | |
3 | 2 | rspcev 3559 | . . 3 ⊢ ((𝑋 ∈ 𝑁 ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
4 | 3 | 3ad2antl3 1180 | . 2 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → ∃𝑥 ∈ 𝑁 𝑥𝐶𝑌) |
5 | simpl1 1184 | . . 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 36223 | . . 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 709 | 1 ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝑁) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 class class class wbr 4966 ‘cfv 6230 Basecbs 16317 ⋖ ccvr 35955 LLinesclln 36184 LPlanesclpl 36185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pr 5226 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3710 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-br 4967 df-opab 5029 df-mpt 5046 df-id 5353 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-iota 6194 df-fun 6232 df-fv 6238 df-lplanes 36192 |
This theorem is referenced by: lplnle 36233 llncvrlpln 36251 lplnexllnN 36257 |
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