| Mathbox for Norm Megill |
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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 39531 | . . 3 ⊢ (𝐾 ∈ 𝐴 → 𝑃 = {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥}) |
| 6 | 5 | eleq2d 2827 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ 𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥})) |
| 7 | breq2 5147 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝑦𝐶𝑥 ↔ 𝑦𝐶𝑋)) | |
| 8 | 7 | rexbidv 3179 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝑁 𝑦𝐶𝑥 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
| 9 | 8 | elrab 3692 | . 2 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐵 ∣ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑥} ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
| 10 | 6, 9 | bitrdi 287 | 1 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 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: islpln4 39533 lplni 39534 lplnbase 39536 lplnnle2at 39543 |
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