Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > islpln4 | Structured version Visualization version GIF version |
Description: The predicate "is a lattice plane". (Contributed by NM, 17-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 |
---|---|
islpln4 | ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lplnset.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lplnset.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
3 | lplnset.n | . . 3 ⊢ 𝑁 = (LLines‘𝐾) | |
4 | lplnset.p | . . 3 ⊢ 𝑃 = (LPlanes‘𝐾) | |
5 | 1, 2, 3, 4 | islpln 33834 | . 2 ⊢ (𝐾 ∈ 𝐴 → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋))) |
6 | 5 | baibd 946 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵) → (𝑋 ∈ 𝑃 ↔ ∃𝑦 ∈ 𝑁 𝑦𝐶𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 ⋖ ccvr 33567 LLinesclln 33795 LPlanesclpl 33796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-lplanes 33803 |
This theorem is referenced by: islpln3 33837 lplncmp 33866 |
Copyright terms: Public domain | W3C validator |