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Theorem poslubmo 16969
 Description: Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
poslubmo.l = (le‘𝐾)
poslubmo.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
poslubmo ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
Distinct variable groups:   𝑥, ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem poslubmo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simplrr 797 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑤𝐵)
2 simprlr 799 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧))
3 simprrl 800 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑤)
4 breq2 4587 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑦 𝑧𝑦 𝑤))
54ralbidv 2969 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑤))
6 breq2 4587 . . . . . . . 8 (𝑧 = 𝑤 → (𝑥 𝑧𝑥 𝑤))
75, 6imbi12d 333 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤)))
87rspcv 3278 . . . . . 6 (𝑤𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) → (∀𝑦𝑆 𝑦 𝑤𝑥 𝑤)))
91, 2, 3, 8syl3c 64 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 𝑤)
10 simplrl 796 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥𝐵)
11 simprrr 801 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))
12 simprll 798 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ∀𝑦𝑆 𝑦 𝑥)
13 breq2 4587 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑦 𝑧𝑦 𝑥))
1413ralbidv 2969 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑦 𝑧 ↔ ∀𝑦𝑆 𝑦 𝑥))
15 breq2 4587 . . . . . . . 8 (𝑧 = 𝑥 → (𝑤 𝑧𝑤 𝑥))
1614, 15imbi12d 333 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑦 𝑧𝑤 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥)))
1716rspcv 3278 . . . . . 6 (𝑥𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧) → (∀𝑦𝑆 𝑦 𝑥𝑤 𝑥)))
1810, 11, 12, 17syl3c 64 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑤 𝑥)
19 poslubmo.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
20 poslubmo.l . . . . . . . . 9 = (le‘𝐾)
2119, 20posasymb 16775 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
22213expb 1258 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2322adantlr 747 . . . . . 6 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2423adantr 480 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
259, 18, 24mpbi2and 958 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))) → 𝑥 = 𝑤)
2625ex 449 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
2726ralrimivva 2954 . 2 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
28 breq2 4587 . . . . 5 (𝑥 = 𝑤 → (𝑦 𝑥𝑦 𝑤))
2928ralbidv 2969 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑦 𝑥 ↔ ∀𝑦𝑆 𝑦 𝑤))
30 breq1 4586 . . . . . 6 (𝑥 = 𝑤 → (𝑥 𝑧𝑤 𝑧))
3130imbi2d 329 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3231ralbidv 2969 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧)))
3329, 32anbi12d 743 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))))
3433rmo4 3366 . 2 (∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ↔ ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)) ∧ (∀𝑦𝑆 𝑦 𝑤 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑤 𝑧))) → 𝑥 = 𝑤))
3527, 34sylibr 223 1 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑦 𝑧𝑥 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃*wrmo 2899   ⊆ wss 3540   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-preset 16751  df-poset 16769 This theorem is referenced by:  poslubd  16971
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