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Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane". (Contributed by NM, 16-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 |
---|---|
islpln | ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lplnset.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lplnset.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
3 | lplnset.n | . . . 4 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | lplnset.p | . . . 4 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | 1, 2, 3, 4 | lplnset 35541 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥}) |
6 | 5 | eleq2d 2862 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})) |
7 | breq2 4845 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝐶𝑥 ↔ 𝑦𝐶𝑋)) | |
8 | 7 | rexbidv 3231 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝑁 𝑦𝐶𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
9 | 8 | elrab 3554 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
10 | 6, 9 | syl6bb 279 | 1 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∃wrex 3088 {crab 3091 class class class wbr 4841 ‘cfv 6099 Basecbs 16180 ⋖ ccvr 35274 LLinesclln 35503 LPlanesclpl 35504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2375 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pr 5095 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-iota 6062 df-fun 6101 df-fv 6107 df-lplanes 35511 |
This theorem is referenced by: islpln4 35543 lplni 35544 lplnbase 35546 lplnnle2at 35553 |
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