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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnbase | Structured version Visualization version GIF version |
Description: A lattice plane is a lattice element. (Contributed by NM, 17-Jun-2012.) |
Ref | Expression |
---|---|
lplnbase.b | ⊢ 𝐵 = (Base‘𝐾) |
lplnbase.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
lplnbase | ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0i 4299 | . . . 4 ⊢ (𝑋 ∈ 𝑃 → ¬ 𝑃 = ∅) | |
2 | lplnbase.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
3 | 2 | eqeq1i 2826 | . . . 4 ⊢ (𝑃 = ∅ ↔ (LPlanes‘𝐾) = ∅) |
4 | 1, 3 | sylnib 330 | . . 3 ⊢ (𝑋 ∈ 𝑃 → ¬ (LPlanes‘𝐾) = ∅) |
5 | fvprc 6663 | . . 3 ⊢ (¬ 𝐾 ∈ V → (LPlanes‘𝐾) = ∅) | |
6 | 4, 5 | nsyl2 143 | . 2 ⊢ (𝑋 ∈ 𝑃 → 𝐾 ∈ V) |
7 | lplnbase.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
8 | eqid 2821 | . . . 4 ⊢ ( ⋖ ‘𝐾) = ( ⋖ ‘𝐾) | |
9 | eqid 2821 | . . . 4 ⊢ (LLines‘𝐾) = (LLines‘𝐾) | |
10 | 7, 8, 9, 2 | islpln 36681 | . . 3 ⊢ (𝐾 ∈ V → (𝑋 ∈ 𝑃 ↔ (𝑋 ∈ 𝐵 ∧ ∃𝑥 ∈ (LLines‘𝐾)𝑥( ⋖ ‘𝐾)𝑋))) |
11 | 10 | simprbda 501 | . 2 ⊢ ((𝐾 ∈ V ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝐵) |
12 | 6, 11 | mpancom 686 | 1 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ∅c0 4291 class class class wbr 5066 ‘cfv 6355 Basecbs 16483 ⋖ ccvr 36413 LLinesclln 36642 LPlanesclpl 36643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-lplanes 36650 |
This theorem is referenced by: islpln2 36687 llnmlplnN 36690 lplnnle2at 36692 lplnneat 36696 lplnnelln 36697 llncvrlpln2 36708 2lplnmN 36710 lplncmp 36713 lplnexatN 36714 lplnexllnN 36715 2llnjaN 36717 islvol3 36727 lvoli3 36728 lvolnle3at 36733 lplncvrlvol2 36766 lplncvrlvol 36767 lvolcmp 36768 2lplnm2N 36772 2lplnmj 36773 dalemyeb 36800 dalem10 36824 dalem16 36830 dalem44 36867 dalem55 36878 |
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