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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 2736 | . . 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 17851 | . 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 17850 | . . . 4 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾))) |
10 | 8, 9 | mpbid 235 | . . 3 ⊢ (𝜑 → {𝑋, 𝑌} ∈ dom (glb‘𝐾)) |
11 | 7, 1, 3, 10 | glbcl 17830 | . 2 ⊢ (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵) |
12 | 6, 11 | eqeltrd 2831 | 1 ⊢ (𝜑 → (𝑋 ∧ 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 {cpr 4529 〈cop 4533 dom cdm 5536 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 glbcglb 17771 meetcmee 17773 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 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 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-glb 17807 df-meet 17809 |
This theorem is referenced by: meetle 17860 latlem 17897 |
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