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 2738 | . . 3 ⊢ (glb‘𝑊) = (glb‘𝑊) | |
3 | clatp0cl.0 | . . 3 ⊢ 0 = (0.‘𝑊) | |
4 | 1, 2, 3 | p0val 17769 | . 2 ⊢ (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵)) |
5 | ssid 3899 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatglbcl 17842 | . . 3 ⊢ ((𝑊 ∈ CLat ∧ 𝐵 ⊆ 𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
7 | 5, 6 | mpan2 691 | . 2 ⊢ (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
8 | 4, 7 | eqeltrd 2833 | 1 ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3843 ‘cfv 6339 Basecbs 16588 glbcglb 17671 0.cp0 17765 CLatccla 17835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-lub 17702 df-glb 17703 df-p0 17767 df-clat 17836 |
This theorem is referenced by: (None) |
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