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Mirrors > Home > MPE Home > Th. List > clatlem | Structured version Visualization version GIF version |
Description: Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.) |
Ref | Expression |
---|---|
clatlem.b | ⊢ 𝐵 = (Base‘𝐾) |
clatlem.u | ⊢ 𝑈 = (lub‘𝐾) |
clatlem.g | ⊢ 𝐺 = (glb‘𝐾) |
Ref | Expression |
---|---|
clatlem | ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clatlem.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | clatlem.u | . . 3 ⊢ 𝑈 = (lub‘𝐾) | |
3 | simpl 486 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat) | |
4 | 1 | fvexi 6753 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4 | elpw2 5255 | . . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
6 | 5 | biimpri 231 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
7 | 6 | adantl 485 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
8 | clatlem.g | . . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾) | |
9 | 1, 2, 8 | isclat 18039 | . . . . . . 7 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
10 | 9 | biimpi 219 | . . . . . 6 ⊢ (𝐾 ∈ CLat → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
11 | 10 | adantr 484 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
12 | 11 | simprld 772 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵) |
13 | 7, 12 | eleqtrrd 2843 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
14 | 1, 2, 3, 13 | lubcl 17896 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
15 | 11 | simprrd 774 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
16 | 7, 15 | eleqtrrd 2843 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
17 | 1, 8, 3, 16 | glbcl 17909 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
18 | 14, 17 | jca 515 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ⊆ wss 3883 𝒫 cpw 4530 dom cdm 5569 ‘cfv 6401 Basecbs 16793 Posetcpo 17847 lubclub 17849 glbcglb 17850 CLatccla 18037 |
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 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-rep 5196 ax-sep 5209 ax-nul 5216 ax-pow 5275 ax-pr 5339 |
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 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3425 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4255 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5153 df-id 5472 df-xp 5575 df-rel 5576 df-cnv 5577 df-co 5578 df-dm 5579 df-rn 5580 df-res 5581 df-ima 5582 df-iota 6359 df-fun 6403 df-fn 6404 df-f 6405 df-f1 6406 df-fo 6407 df-f1o 6408 df-fv 6409 df-riota 7192 df-lub 17885 df-glb 17886 df-clat 18038 |
This theorem is referenced by: clatlubcl 18042 clatglbcl 18044 |
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