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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > clatp1cl | Structured version Visualization version GIF version |
Description: The poset one of a complete lattice belongs to its base. (Contributed by Thierry Arnoux, 17-Feb-2018.) |
Ref | Expression |
---|---|
clatp1cl.b | ⊢ 𝐵 = (Base‘𝑊) |
clatp1cl.1 | ⊢ 1 = (1.‘𝑊) |
Ref | Expression |
---|---|
clatp1cl | ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatp1cl.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | eqid 2728 | . . 3 ⊢ (lub‘𝑊) = (lub‘𝑊) | |
3 | clatp1cl.1 | . . 3 ⊢ 1 = (1.‘𝑊) | |
4 | 1, 2, 3 | p1val 18413 | . 2 ⊢ (𝑊 ∈ CLat → 1 = ((lub‘𝑊)‘𝐵)) |
5 | ssid 4000 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatlubcl 18488 | . . 3 ⊢ ((𝑊 ∈ CLat ∧ 𝐵 ⊆ 𝐵) → ((lub‘𝑊)‘𝐵) ∈ 𝐵) |
7 | 5, 6 | mpan2 690 | . 2 ⊢ (𝑊 ∈ CLat → ((lub‘𝑊)‘𝐵) ∈ 𝐵) |
8 | 4, 7 | eqeltrd 2829 | 1 ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3945 ‘cfv 6542 Basecbs 17173 lubclub 18294 1.cp1 18409 CLatccla 18483 |
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 2167 ax-ext 2699 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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 18331 df-glb 18332 df-p1 18411 df-clat 18484 |
This theorem is referenced by: (None) |
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