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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 2821 | . . . 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 17622 | . . 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 263 | . . . . . 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 483 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉) |
12 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾)) | |
13 | 8, 9, 2, 10, 11, 12 | glbeu 17600 | . . . . 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 415 | . . . 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 17626 | . . . . . 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 586 | . . . . 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 3389 | . . . 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 241 | . . 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 242 | . 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 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃!wreu 3140 {cpr 4562 〈cop 4566 class class class wbr 5058 dom cdm 5549 ‘cfv 6349 Basecbs 16477 lecple 16566 glbcglb 17547 meetcmee 17549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-oprab 7154 df-glb 17579 df-meet 17581 |
This theorem is referenced by: meetlem 17629 |
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