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 2821 | . . 3 ⊢ (lub‘𝑊) = (lub‘𝑊) | |
3 | clatp1cl.1 | . . 3 ⊢ 1 = (1.‘𝑊) | |
4 | 1, 2, 3 | p1val 17651 | . 2 ⊢ (𝑊 ∈ CLat → 1 = ((lub‘𝑊)‘𝐵)) |
5 | ssid 3988 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatlubcl 17721 | . . 3 ⊢ ((𝑊 ∈ CLat ∧ 𝐵 ⊆ 𝐵) → ((lub‘𝑊)‘𝐵) ∈ 𝐵) |
7 | 5, 6 | mpan2 689 | . 2 ⊢ (𝑊 ∈ CLat → ((lub‘𝑊)‘𝐵) ∈ 𝐵) |
8 | 4, 7 | eqeltrd 2913 | 1 ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6354 Basecbs 16482 lubclub 17551 1.cp1 17647 CLatccla 17716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-lub 17583 df-glb 17584 df-p1 17649 df-clat 17717 |
This theorem is referenced by: (None) |
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