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Mirrors > Home > MPE Home > Th. List > Mathboxes > posjidm | Structured version Visualization version GIF version |
Description: Poset join is idempotent. latjidm 17922 could be shortened by this. (Contributed by Zhi Wang, 27-Sep-2024.) |
Ref | Expression |
---|---|
posjidm.b | ⊢ 𝐵 = (Base‘𝐾) |
posjidm.j | ⊢ ∨ = (join‘𝐾) |
Ref | Expression |
---|---|
posjidm | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
2 | posjidm.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
3 | simpl 486 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝐾 ∈ Poset) | |
4 | simpr 488 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
5 | 1, 2, 3, 4, 4 | joinval 17837 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = ((lub‘𝐾)‘{𝑋, 𝑋})) |
6 | posjidm.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
7 | eqid 2736 | . . 3 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 6, 7 | posref 17779 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾)𝑋) |
9 | eqidd 2737 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → {𝑋, 𝑋} = {𝑋, 𝑋}) | |
10 | 3, 6, 4, 4, 7, 8, 9, 1 | lubpr 45874 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → ((lub‘𝐾)‘{𝑋, 𝑋}) = 𝑋) |
11 | 5, 10 | eqtrd 2771 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 {cpr 4529 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 lecple 16756 Posetcpo 17768 lubclub 17770 joincjn 17772 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-proset 17756 df-poset 17774 df-lub 17806 df-join 17808 |
This theorem is referenced by: (None) |
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