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Theorem posglbmo 16970
 Description: Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.)
Hypotheses
Ref Expression
poslubmo.l = (le‘𝐾)
poslubmo.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
posglbmo ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
Distinct variable groups:   𝑥, ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧

Proof of Theorem posglbmo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 simplrr 797 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑤𝐵)
2 simprlr 799 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥))
3 simprrl 800 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑦𝑆 𝑤 𝑦)
4 breq1 4586 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 𝑦𝑤 𝑦))
54ralbidv 2969 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
6 breq1 4586 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧 𝑥𝑤 𝑥))
75, 6imbi12d 333 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥)))
87rspcv 3278 . . . . . 6 (𝑤𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) → (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥)))
91, 2, 3, 8syl3c 64 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑤 𝑥)
10 simplrl 796 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥𝐵)
11 simprrr 801 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))
12 simprll 798 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑦𝑆 𝑥 𝑦)
13 breq1 4586 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 𝑦𝑥 𝑦))
1413ralbidv 2969 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑥 𝑦))
15 breq1 4586 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 𝑤𝑥 𝑤))
1614, 15imbi12d 333 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑤) ↔ (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤)))
1716rspcv 3278 . . . . . 6 (𝑥𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤) → (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤)))
1810, 11, 12, 17syl3c 64 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥 𝑤)
19 ancom 465 . . . . . . . . 9 ((𝑤 𝑥𝑥 𝑤) ↔ (𝑥 𝑤𝑤 𝑥))
20 poslubmo.b . . . . . . . . . 10 𝐵 = (Base‘𝐾)
21 poslubmo.l . . . . . . . . . 10 = (le‘𝐾)
2220, 21posasymb 16775 . . . . . . . . 9 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2319, 22syl5bb 271 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
24233expb 1258 . . . . . . 7 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
2524adantlr 747 . . . . . 6 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
2625adantr 480 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
279, 18, 26mpbi2and 958 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥 = 𝑤)
2827ex 449 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
2928ralrimivva 2954 . 2 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
30 breq1 4586 . . . . 5 (𝑥 = 𝑤 → (𝑥 𝑦𝑤 𝑦))
3130ralbidv 2969 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑥 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
32 breq2 4587 . . . . . 6 (𝑥 = 𝑤 → (𝑧 𝑥𝑧 𝑤))
3332imbi2d 329 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3433ralbidv 2969 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3531, 34anbi12d 743 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))))
3635rmo4 3366 . 2 (∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
3729, 36sylibr 223 1 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = 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: (None)
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