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Theorem posglbmo 17745
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 simprrl 777 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑦𝑆 𝑤 𝑦)
2 breq1 5060 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 𝑦𝑤 𝑦))
32ralbidv 3194 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
4 breq1 5060 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧 𝑥𝑤 𝑥))
53, 4imbi12d 346 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥)))
6 simprlr 776 . . . . . . 7 ((((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))
7 simplrr 774 . . . . . . 7 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑤𝐵)
85, 6, 7rspcdva 3622 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥))
91, 8mpd 15 . . . . 5 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑤 𝑥)
10 simprll 775 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → ∀𝑦𝑆 𝑥 𝑦)
11 breq1 5060 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 𝑦𝑥 𝑦))
1211ralbidv 3194 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑥 𝑦))
13 breq1 5060 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 𝑤𝑥 𝑤))
1412, 13imbi12d 346 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑤) ↔ (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤)))
15 simprrr 778 . . . . . . 7 ((((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))
16 simplrl 773 . . . . . . 7 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥𝐵)
1714, 15, 16rspcdva 3622 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤))
1810, 17mpd 15 . . . . 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 461 . . . . . . . 8 ((𝑤 𝑥𝑥 𝑤) ↔ (𝑥 𝑤𝑤 𝑥))
20 poslubmo.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
21 poslubmo.l . . . . . . . . 9 = (le‘𝐾)
2220, 21posasymb 17550 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2319, 22syl5bb 284 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
24233expb 1112 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
2524ad4ant13 747 . . . . 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))
269, 18, 25mpbi2and 708 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥 = 𝑤)
2726ex 413 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
2827ralrimivva 3188 . 2 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∀𝑥𝐵𝑤𝐵 (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
29 breq1 5060 . . . . 5 (𝑥 = 𝑤 → (𝑥 𝑦𝑤 𝑦))
3029ralbidv 3194 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑥 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
31 breq2 5061 . . . . . 6 (𝑥 = 𝑤 → (𝑧 𝑥𝑧 𝑤))
3231imbi2d 342 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3332ralbidv 3194 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3430, 33anbi12d 630 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))))
3534rmo4 3718 . 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))
3628, 35sylibr 235 1 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ∃*wrmo 3138  wss 3933   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  Posetcpo 17538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-proset 17526  df-poset 17544
This theorem is referenced by: (None)
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