Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2824 | . . 3 ⊢ (lub‘𝑊) = (lub‘𝑊) | |
3 | clatp1cl.1 | . . 3 ⊢ 1 = (1.‘𝑊) | |
4 | 1, 2, 3 | p1val 17655 | . 2 ⊢ (𝑊 ∈ CLat → 1 = ((lub‘𝑊)‘𝐵)) |
5 | ssid 3992 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
6 | 1, 2 | clatlubcl 17725 | . . 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 2916 | 1 ⊢ (𝑊 ∈ CLat → 1 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ⊆ wss 3939 ‘cfv 6358 Basecbs 16486 lubclub 17555 1.cp1 17651 CLatccla 17720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-lub 17587 df-glb 17588 df-p1 17653 df-clat 17721 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |