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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lhpelim | Structured version Visualization version GIF version |
Description: Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 35838 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.) |
Ref | Expression |
---|---|
lhpelim.b | ⊢ 𝐵 = (Base‘𝐾) |
lhpelim.l | ⊢ ≤ = (le‘𝐾) |
lhpelim.j | ⊢ ∨ = (join‘𝐾) |
lhpelim.m | ⊢ ∧ = (meet‘𝐾) |
lhpelim.a | ⊢ 𝐴 = (Atoms‘𝐾) |
lhpelim.h | ⊢ 𝐻 = (LHyp‘𝐾) |
Ref | Expression |
---|---|
lhpelim | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊) = (𝑋 ∧ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lhpelim.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
2 | lhpelim.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
3 | eqid 2771 | . . . . 5 ⊢ (0.‘𝐾) = (0.‘𝐾) | |
4 | lhpelim.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | lhpelim.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | 1, 2, 3, 4, 5 | lhpmat 35838 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
7 | 6 | 3adant3 1126 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → (𝑃 ∧ 𝑊) = (0.‘𝐾)) |
8 | 7 | oveq1d 6808 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑊) ∨ (𝑋 ∧ 𝑊)) = ((0.‘𝐾) ∨ (𝑋 ∧ 𝑊))) |
9 | simp1l 1239 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ HL) | |
10 | simp2l 1241 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝑃 ∈ 𝐴) | |
11 | hllat 35172 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
12 | 9, 11 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
13 | simp3 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
14 | simp1r 1240 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ 𝐻) | |
15 | lhpelim.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
16 | 15, 5 | lhpbase 35806 | . . . . 5 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵) |
17 | 14, 16 | syl 17 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝑊 ∈ 𝐵) |
18 | 15, 2 | latmcl 17260 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵) |
19 | 12, 13, 17, 18 | syl3anc 1476 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵) |
20 | 15, 1, 2 | latmle2 17285 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊) |
21 | 12, 13, 17, 20 | syl3anc 1476 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊) |
22 | lhpelim.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
23 | 15, 1, 22, 2, 4 | atmod4i2 35675 | . . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ (𝑋 ∧ 𝑊) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) ∧ (𝑋 ∧ 𝑊) ≤ 𝑊) → ((𝑃 ∧ 𝑊) ∨ (𝑋 ∧ 𝑊)) = ((𝑃 ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊)) |
24 | 9, 10, 19, 17, 21, 23 | syl131anc 1489 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∧ 𝑊) ∨ (𝑋 ∧ 𝑊)) = ((𝑃 ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊)) |
25 | hlol 35170 | . . . 4 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OL) | |
26 | 9, 25 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OL) |
27 | 15, 22, 3 | olj02 35035 | . . 3 ⊢ ((𝐾 ∈ OL ∧ (𝑋 ∧ 𝑊) ∈ 𝐵) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ 𝑊)) |
28 | 26, 19, 27 | syl2anc 573 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → ((0.‘𝐾) ∨ (𝑋 ∧ 𝑊)) = (𝑋 ∧ 𝑊)) |
29 | 8, 24, 28 | 3eqtr3d 2813 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) → ((𝑃 ∨ (𝑋 ∧ 𝑊)) ∧ 𝑊) = (𝑋 ∧ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 class class class wbr 4786 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 lecple 16156 joincjn 17152 meetcmee 17153 0.cp0 17245 Latclat 17253 OLcol 34983 Atomscatm 35072 HLchlt 35159 LHypclh 35792 |
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 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
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 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 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 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 |
This theorem is referenced by: cdleme48b 36312 cdlemg7fvN 36433 |
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