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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 29494 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 | dfsn2 4554 | . . . 4 ⊢ {𝑋} = {𝑋, 𝑋} | |
2 | 1 | fveq2i 6720 | . . 3 ⊢ (𝑈‘{𝑋}) = (𝑈‘{𝑋, 𝑋}) |
3 | lubsn.u | . . . 4 ⊢ 𝑈 = (lub‘𝐾) | |
4 | eqid 2737 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
5 | simpl 486 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) | |
6 | simpr 488 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
7 | 3, 4, 5, 6, 6 | joinval 17883 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = (𝑈‘{𝑋, 𝑋})) |
8 | 2, 7 | eqtr4id 2797 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = (𝑋(join‘𝐾)𝑋)) |
9 | lubsn.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 9, 4 | latjidm 17968 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = 𝑋) |
11 | 8, 10 | eqtrd 2777 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 {csn 4541 {cpr 4543 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 lubclub 17816 joincjn 17818 Latclat 17937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-proset 17802 df-poset 17820 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-lat 17938 |
This theorem is referenced by: lubel 18020 |
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