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Mirrors > Home > MPE Home > Th. List > joineu | Structured version Visualization version GIF version |
Description: Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018.) |
Ref | Expression |
---|---|
joinval2.b | ⊢ 𝐵 = (Base‘𝐾) |
joinval2.l | ⊢ ≤ = (le‘𝐾) |
joinval2.j | ⊢ ∨ = (join‘𝐾) |
joinval2.k | ⊢ (𝜑 → 𝐾 ∈ 𝑉) |
joinval2.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
joinval2.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
joinlem.e | ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) |
Ref | Expression |
---|---|
joineu | ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | joinlem.e | . 2 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom ∨ ) | |
2 | eqid 2823 | . . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾) | |
3 | joinval2.j | . . . 4 ⊢ ∨ = (join‘𝐾) | |
4 | joinval2.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉) | |
5 | joinval2.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | joinval2.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 2, 3, 4, 5, 6 | joindef 17616 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ ↔ {𝑋, 𝑌} ∈ dom (lub‘𝐾))) |
8 | joinval2.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
9 | joinval2.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
10 | biid 263 | . . . . . 6 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) | |
11 | 4 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉) |
12 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → {𝑋, 𝑌} ∈ dom (lub‘𝐾)) | |
13 | 8, 9, 2, 10, 11, 12 | lubeu 17595 | . . . . 5 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) |
14 | 13 | ex 415 | . . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))) |
15 | 8, 9, 3, 4, 5, 6 | joinval2lem 17620 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
16 | 5, 6, 15 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
17 | 16 | reubidv 3391 | . . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
18 | 14, 17 | sylibd 241 | . . 3 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
19 | 7, 18 | sylbid 242 | . 2 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ dom ∨ → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))) |
20 | 1, 19 | mpd 15 | 1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃!wreu 3142 {cpr 4571 〈cop 4575 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 Basecbs 16485 lecple 16574 lubclub 17554 joincjn 17556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-oprab 7162 df-lub 17586 df-join 17588 |
This theorem is referenced by: joinlem 17623 |
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