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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 29182 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 2819 | . . . 4 ⊢ (join‘𝐾) = (join‘𝐾) | |
3 | simpl 485 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Lat) | |
4 | simpr 487 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | 1, 2, 3, 4, 4 | joinval 17607 | . . 3 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = (𝑈‘{𝑋, 𝑋})) |
6 | dfsn2 4572 | . . . 4 ⊢ {𝑋} = {𝑋, 𝑋} | |
7 | 6 | fveq2i 6666 | . . 3 ⊢ (𝑈‘{𝑋}) = (𝑈‘{𝑋, 𝑋}) |
8 | 5, 7 | syl6reqr 2873 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = (𝑋(join‘𝐾)𝑋)) |
9 | lubsn.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
10 | 9, 2 | latjidm 17676 | . 2 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋(join‘𝐾)𝑋) = 𝑋) |
11 | 8, 10 | eqtrd 2854 | 1 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 {csn 4559 {cpr 4561 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 lubclub 17544 joincjn 17546 Latclat 17647 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-proset 17530 df-poset 17548 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-lat 17648 |
This theorem is referenced by: lubel 17724 |
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