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Mirrors > Home > MPE Home > Th. List > lubid | Structured version Visualization version GIF version |
Description: The LUB of elements less than or equal to a fixed value equals that value. (Contributed by NM, 19-Oct-2011.) (Revised by NM, 7-Sep-2018.) |
Ref | Expression |
---|---|
lubid.b | ⊢ 𝐵 = (Base‘𝐾) |
lubid.l | ⊢ ≤ = (le‘𝐾) |
lubid.u | ⊢ 𝑈 = (lub‘𝐾) |
lubid.k | ⊢ (𝜑 → 𝐾 ∈ Poset) |
lubid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
lubid | ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubid.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | lubid.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
3 | lubid.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
4 | biid 263 | . . 3 ⊢ ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤))) | |
5 | lubid.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Poset) | |
6 | ssrab2 4039 | . . . 4 ⊢ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵 | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋} ⊆ 𝐵) |
8 | 1, 2, 3, 4, 5, 7 | lubval 17572 | . 2 ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = (℩𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)))) |
9 | lubid.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
10 | 1, 2, 3, 5, 9 | lublecllem 17576 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤)) ↔ 𝑥 = 𝑋)) |
11 | 9, 10 | riota5 7124 | . 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑥 ∧ ∀𝑤 ∈ 𝐵 (∀𝑧 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}𝑧 ≤ 𝑤 → 𝑥 ≤ 𝑤))) = 𝑋) |
12 | 8, 11 | eqtrd 2855 | 1 ⊢ (𝜑 → (𝑈‘{𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑋}) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3133 {crab 3137 ⊆ wss 3919 class class class wbr 5047 ‘cfv 6336 ℩crio 7094 Basecbs 16461 lecple 16550 Posetcpo 17528 lubclub 17530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 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 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-proset 17516 df-poset 17534 df-lub 17562 |
This theorem is referenced by: atlatmstc 36482 |
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