Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clatp0cl | Structured version Visualization version GIF version |
Description: The poset zero of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
Ref | Expression |
---|---|
clatp0cl.b | ⊢ 𝐵 = (Base‘𝑊) |
clatp0cl.0 | ⊢ 0 = (0.‘𝑊) |
Ref | Expression |
---|---|
clatp0cl | ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatp0cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | eqid 2818 | . . 3 ⊢ (glb‘𝑊) = (glb‘𝑊) | |
3 | clatp0cl.0 | . . 3 ⊢ 0 = (0.‘𝑊) | |
4 | 1, 2, 3 | p0val 17639 | . 2 ⊢ (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵)) |
5 | ssid 3986 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatglbcl 17712 | . . 3 ⊢ ((𝑊 ∈ CLat ∧ 𝐵 ⊆ 𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
7 | 5, 6 | mpan2 687 | . 2 ⊢ (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
8 | 4, 7 | eqeltrd 2910 | 1 ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 Basecbs 16471 glbcglb 17541 0.cp0 17635 CLatccla 17705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-lub 17572 df-glb 17573 df-p0 17637 df-clat 17706 |
This theorem is referenced by: (None) |
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