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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 2733 | . . 3 ⊢ (glb‘𝑊) = (glb‘𝑊) | |
3 | clatp0cl.0 | . . 3 ⊢ 0 = (0.‘𝑊) | |
4 | 1, 2, 3 | p0val 18474 | . 2 ⊢ (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵)) |
5 | ssid 4018 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatglbcl 18552 | . . 3 ⊢ ((𝑊 ∈ CLat ∧ 𝐵 ⊆ 𝐵) → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
7 | 5, 6 | mpan2 690 | . 2 ⊢ (𝑊 ∈ CLat → ((glb‘𝑊)‘𝐵) ∈ 𝐵) |
8 | 4, 7 | eqeltrd 2837 | 1 ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ⊆ wss 3963 ‘cfv 6559 Basecbs 17235 glbcglb 18357 0.cp0 18470 CLatccla 18545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4916 df-iun 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6511 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7382 df-lub 18393 df-glb 18394 df-p0 18472 df-clat 18546 |
This theorem is referenced by: (None) |
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