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Mirrors > Home > MPE Home > Th. List > lubsn | Structured version Visualization version GIF version |
Description: The least upper bound of a singleton. (chsupsn 28606 analog.) (Contributed by NM, 20-Oct-2011.) |
Ref | Expression |
---|---|
lubsn.b | ⊢ 𝐵 = (Base‘𝐾) |
lubsn.u | ⊢ 𝑈 = (lub‘𝐾) |
Ref | Expression |
---|---|
lubsn | ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubsn.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
2 | eqid 2770 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | simpl 468 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) | |
4 | simpr 471 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | 1, 2, 3, 4, 4 | joinval 17212 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = (𝑈‘{𝑋, 𝑋})) |
6 | dfsn2 4327 | . . . 4 ⊢ {𝑋} = {𝑋, 𝑋} | |
7 | 6 | fveq2i 6335 | . . 3 ⊢ (𝑈‘{𝑋}) = (𝑈‘{𝑋, 𝑋}) |
8 | 5, 7 | syl6reqr 2823 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = (𝑋(join‘𝐾)𝑋)) |
9 | lubsn.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 9, 2 | latjidm 17281 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = 𝑋) |
11 | 8, 10 | eqtrd 2804 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 {csn 4314 {cpr 4316 ‘cfv 6031 (class class class)co 6792 Basecbs 16063 lubclub 17149 joincjn 17151 Latclat 17252 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-preset 17135 df-poset 17153 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-lat 17253 |
This theorem is referenced by: lubel 17329 |
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