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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > olm02 | Structured version Visualization version GIF version |
Description: Meet with lattice zero is zero. (Contributed by NM, 9-Oct-2012.) |
Ref | Expression |
---|---|
olm0.b | ⊢ 𝐵 = (Base‘𝐾) |
olm0.m | ⊢ ∧ = (meet‘𝐾) |
olm0.z | ⊢ 0 = (0.‘𝐾) |
Ref | Expression |
---|---|
olm02 | ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∧ 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ollat 35288 | . . . 4 ⊢ (𝐾 ∈ OL → 𝐾 ∈ Lat) | |
2 | 1 | adantr 474 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) |
3 | simpr 479 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
4 | olop 35289 | . . . . 5 ⊢ (𝐾 ∈ OL → 𝐾 ∈ OP) | |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ OP) |
6 | olm0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
7 | olm0.z | . . . . 5 ⊢ 0 = (0.‘𝐾) | |
8 | 6, 7 | op0cl 35259 | . . . 4 ⊢ (𝐾 ∈ OP → 0 ∈ 𝐵) |
9 | 5, 8 | syl 17 | . . 3 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
10 | olm0.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
11 | 6, 10 | latmcom 17428 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵) → (𝑋 ∧ 0 ) = ( 0 ∧ 𝑋)) |
12 | 2, 3, 9, 11 | syl3anc 1496 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = ( 0 ∧ 𝑋)) |
13 | 6, 10, 7 | olm01 35311 | . 2 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 0 ) = 0 ) |
14 | 12, 13 | eqtr3d 2863 | 1 ⊢ ((𝐾 ∈ OL ∧ 𝑋 ∈ 𝐵) → ( 0 ∧ 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 meetcmee 17298 0.cp0 17390 Latclat 17398 OPcops 35247 OLcol 35249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-proset 17281 df-poset 17299 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-lat 17399 df-oposet 35251 df-ol 35253 |
This theorem is referenced by: cdleme15b 36350 |
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