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| Mirrors > Home > MPE Home > Th. List > meetcl | Structured version Visualization version GIF version | ||
| Description: Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.) |
| Ref | Expression |
|---|---|
| meetcl.b | ⊢ 𝐵 = (Base‘𝐾) |
| meetcl.m | ⊢ ∧ = (meet‘𝐾) |
| meetcl.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
| meetcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| meetcl.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| meetcl.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
| Ref | Expression |
|---|---|
| meetcl | ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
| 2 | meetcl.m | . . 3 ⊢ ∧ = (meet‘𝐾) | |
| 3 | meetcl.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
| 4 | meetcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 5 | meetcl.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 6 | 1, 2, 3, 4, 5 | meetval 18436 | . 2 ⊢ (𝜑 → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) |
| 7 | meetcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
| 8 | meetcl.e | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) | |
| 9 | 1, 2, 3, 4, 5 | meetdef 18435 | . . . 4 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾))) |
| 10 | 8, 9 | mpbid 232 | . . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ dom (glb‘𝐾)) |
| 11 | 7, 1, 3, 10 | glbcl 18415 | . 2 ⊢ (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵) |
| 12 | 6, 11 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 {cpr 4628 〈cop 4632 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 glbcglb 18356 meetcmee 18358 |
| 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 ax-un 7755 |
| 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-ov 7434 df-oprab 7435 df-glb 18392 df-meet 18394 |
| This theorem is referenced by: meetle 18445 latlem 18482 |
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