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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 2727 | . . 3 ⊢ (glb‘𝑊) = (glb‘𝑊) | |
3 | clatp0cl.0 | . . 3 ⊢ 0 = (0.‘𝑊) | |
4 | 1, 2, 3 | p0val 18410 | . 2 ⊢ (𝑊 ∈ CLat → 0 = ((glb‘𝑊)‘𝐵)) |
5 | ssid 4000 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatglbcl 18488 | . . 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 2828 | 1 ⊢ (𝑊 ∈ CLat → 0 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ‘cfv 6542 Basecbs 17171 glbcglb 18293 0.cp0 18406 CLatccla 18481 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-lub 18329 df-glb 18330 df-p0 18408 df-clat 18482 |
This theorem is referenced by: (None) |
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