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| Description: Properties that determine the greatest lower bound of a complete lattice. (Contributed by Mario Carneiro, 19-Mar-2014.) | 
| Ref | Expression | 
|---|---|
| isglbd.b | ⊢ 𝐵 = (Base‘𝐾) | 
| isglbd.l | ⊢ ≤ = (le‘𝐾) | 
| isglbd.g | ⊢ 𝐺 = (glb‘𝐾) | 
| isglbd.1 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦) | 
| isglbd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻) | 
| isglbd.3 | ⊢ (𝜑 → 𝐾 ∈ CLat) | 
| isglbd.4 | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | 
| isglbd.5 | ⊢ (𝜑 → 𝐻 ∈ 𝐵) | 
| Ref | Expression | 
|---|---|
| isglbd | ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | isglbd.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | isglbd.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
| 3 | isglbd.g | . . 3 ⊢ 𝐺 = (glb‘𝐾) | |
| 4 | biid 261 | . . 3 ⊢ ((∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)) ↔ (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) | |
| 5 | isglbd.3 | . . 3 ⊢ (𝜑 → 𝐾 ∈ CLat) | |
| 6 | isglbd.4 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
| 7 | 1, 2, 3, 4, 5, 6 | glbval 18414 | . 2 ⊢ (𝜑 → (𝐺‘𝑆) = (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)))) | 
| 8 | isglbd.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝐻 ≤ 𝑦) | |
| 9 | 8 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦) | 
| 10 | isglbd.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑥 ≤ 𝐻) | |
| 11 | 10 | 3exp 1120 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))) | 
| 12 | 11 | ralrimiv 3145 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)) | 
| 13 | isglbd.5 | . . . 4 ⊢ (𝜑 → 𝐻 ∈ 𝐵) | |
| 14 | 1, 3 | clatglbcl2 18551 | . . . . . 6 ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → 𝑆 ∈ dom 𝐺) | 
| 15 | 5, 6, 14 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝐺) | 
| 16 | 1, 2, 3, 4, 5, 15 | glbeu 18413 | . . . 4 ⊢ (𝜑 → ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) | 
| 17 | breq1 5146 | . . . . . . 7 ⊢ (ℎ = 𝐻 → (ℎ ≤ 𝑦 ↔ 𝐻 ≤ 𝑦)) | |
| 18 | 17 | ralbidv 3178 | . . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦)) | 
| 19 | breq2 5147 | . . . . . . . 8 ⊢ (ℎ = 𝐻 → (𝑥 ≤ ℎ ↔ 𝑥 ≤ 𝐻)) | |
| 20 | 19 | imbi2d 340 | . . . . . . 7 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))) | 
| 21 | 20 | ralbidv 3178 | . . . . . 6 ⊢ (ℎ = 𝐻 → (∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ) ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻))) | 
| 22 | 18, 21 | anbi12d 632 | . . . . 5 ⊢ (ℎ = 𝐻 → ((∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ)) ↔ (∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)))) | 
| 23 | 22 | riota2 7413 | . . . 4 ⊢ ((𝐻 ∈ 𝐵 ∧ ∃!ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) → ((∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)) ↔ (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)) | 
| 24 | 13, 16, 23 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝐻 ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ 𝐻)) ↔ (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻)) | 
| 25 | 9, 12, 24 | mpbi2and 712 | . 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐵 (∀𝑦 ∈ 𝑆 ℎ ≤ 𝑦 ∧ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 → 𝑥 ≤ ℎ))) = 𝐻) | 
| 26 | 7, 25 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝐺‘𝑆) = 𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃!wreu 3378 ⊆ wss 3951 class class class wbr 5143 dom cdm 5685 ‘cfv 6561 ℩crio 7387 Basecbs 17247 lecple 17304 glbcglb 18356 CLatccla 18543 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-glb 18392 df-clat 18544 | 
| This theorem is referenced by: dihglblem2N 41296 | 
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