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Mirrors > Home > MPE Home > Th. List > meeteu | Structured version Visualization version GIF version |
Description: Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
meetval2.b | ⊢ 𝐵 = (Base‘𝐾) |
meetval2.l | ⊢ ≤ = (le‘𝐾) |
meetval2.m | ⊢ ∧ = (meet‘𝐾) |
meetval2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
meetval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
meetval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
meetlem.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) |
Ref | Expression |
---|---|
meeteu | ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | meetlem.e | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∧ ) | |
2 | eqid 2778 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
3 | meetval2.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
4 | meetval2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | meetval2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | meetval2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 2, 3, 4, 5, 6 | meetdef 17415 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾))) |
8 | meetval2.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
9 | meetval2.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
10 | biid 253 | . . . . . 6 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) | |
11 | 4 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉) |
12 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾)) | |
13 | 8, 9, 2, 10, 11, 12 | glbeu 17393 | . . . . 5 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))) |
14 | 13 | ex 403 | . . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))) |
15 | 8, 9, 3, 4, 5, 6 | meetval2lem 17419 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
16 | 5, 6, 15 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
17 | 16 | reubidv 3314 | . . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
18 | 14, 17 | sylibd 231 | . . 3 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
19 | 7, 18 | sylbid 232 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∧ → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))) |
20 | 1, 19 | mpd 15 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∃!wreu 3092 {cpr 4400 〈cop 4404 class class class wbr 4888 dom cdm 5357 ‘cfv 6137 Basecbs 16266 lecple 16356 glbcglb 17340 meetcmee 17342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-oprab 6928 df-glb 17372 df-meet 17374 |
This theorem is referenced by: meetlem 17422 |
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