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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 485 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝐾 ∈ CLat) | |
4 | 1 | fvexi 6686 | . . . . . . 7 ⊢ 𝐵 ∈ V |
5 | 4 | elpw2 5250 | . . . . . 6 ⊢ (𝑆 ∈ 𝒫 𝐵 ↔ 𝑆 ⊆ 𝐵) |
6 | 5 | biimpri 230 | . . . . 5 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ 𝒫 𝐵) |
7 | 6 | adantl 484 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
8 | clatlem.g | . . . . . . . 8 ⊢ 𝐺 = (glb‘𝐾) | |
9 | 1, 2, 8 | isclat 17721 | . . . . . . 7 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
10 | 9 | biimpi 218 | . . . . . 6 ⊢ (𝐾 ∈ CLat → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
11 | 10 | adantr 483 | . . . . 5 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵))) |
12 | 11 | simprld 770 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝑈 = 𝒫 𝐵) |
13 | 7, 12 | eleqtrrd 2918 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝑈) |
14 | 1, 2, 3, 13 | lubcl 17597 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝑈‘𝑆) ∈ 𝐵) |
15 | 11 | simprrd 772 | . . . 4 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → dom 𝐺 = 𝒫 𝐵) |
16 | 7, 15 | eleqtrrd 2918 | . . 3 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) |
17 | 1, 8, 3, 16 | glbcl 17610 | . 2 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → (𝐺‘𝑆) ∈ 𝐵) |
18 | 14, 17 | jca 514 | 1 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 𝒫 cpw 4541 dom cdm 5557 ‘cfv 6357 Basecbs 16485 Posetcpo 17552 lubclub 17554 glbcglb 17555 CLatccla 17719 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
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 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-lub 17586 df-glb 17587 df-clat 17720 |
This theorem is referenced by: clatlubcl 17724 clatglbcl 17726 |
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