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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lplnnlt | Structured version Visualization version GIF version |
Description: Two lattice planes cannot satisfy the less than relation. (Contributed by NM, 7-Jul-2012.) |
Ref | Expression |
---|---|
lplnnlt.s | ⊢ < = (lt‘𝐾) |
lplnnlt.p | ⊢ 𝑃 = (LPlanes‘𝐾) |
Ref | Expression |
---|---|
lplnnlt | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 < 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lplnnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
2 | 1 | pltirr 17171 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃) → ¬ 𝑋 < 𝑋) |
3 | 2 | 3adant3 1126 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 < 𝑋) |
4 | breq2 4791 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 < 𝑋 ↔ 𝑋 < 𝑌)) | |
5 | 4 | notbid 307 | . . 3 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 < 𝑋 ↔ ¬ 𝑋 < 𝑌)) |
6 | 3, 5 | syl5ibcom 235 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 = 𝑌 → ¬ 𝑋 < 𝑌)) |
7 | eqid 2771 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 17169 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
9 | lplnnlt.p | . . . . 5 ⊢ 𝑃 = (LPlanes‘𝐾) | |
10 | 7, 9 | lplncmp 35369 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋(le‘𝐾)𝑌 ↔ 𝑋 = 𝑌)) |
11 | 8, 10 | sylibd 229 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 < 𝑌 → 𝑋 = 𝑌)) |
12 | 11 | necon3ad 2956 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → (𝑋 ≠ 𝑌 → ¬ 𝑋 < 𝑌)) |
13 | 6, 12 | pm2.61dne 3029 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑃 ∧ 𝑌 ∈ 𝑃) → ¬ 𝑋 < 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 class class class wbr 4787 ‘cfv 6030 lecple 16156 ltcplt 17149 HLchlt 35157 LPlanesclpl 35299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6757 df-ov 6799 df-oprab 6800 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-lat 17254 df-clat 17316 df-oposet 34983 df-ol 34985 df-oml 34986 df-covers 35073 df-ats 35074 df-atl 35105 df-cvlat 35129 df-hlat 35158 df-llines 35305 df-lplanes 35306 |
This theorem is referenced by: lvolnle3at 35389 |
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