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Theorem posglbmo 17344
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 820 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑤𝐵)
2 simprlr 822 . . . . . 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 823 . . . . . 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 4803 . . . . . . . . 9 (𝑧 = 𝑤 → (𝑧 𝑦𝑤 𝑦))
54ralbidv 3120 . . . . . . . 8 (𝑧 = 𝑤 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
6 breq1 4803 . . . . . . . 8 (𝑧 = 𝑤 → (𝑧 𝑥𝑤 𝑥))
75, 6imbi12d 333 . . . . . . 7 (𝑧 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥)))
87rspcv 3441 . . . . . 6 (𝑤𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) → (∀𝑦𝑆 𝑤 𝑦𝑤 𝑥)))
91, 2, 3, 8syl3c 66 . . . . 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 819 . . . . . 6 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥𝐵)
11 simprrr 824 . . . . . 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 821 . . . . . 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 4803 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧 𝑦𝑥 𝑦))
1413ralbidv 3120 . . . . . . . 8 (𝑧 = 𝑥 → (∀𝑦𝑆 𝑧 𝑦 ↔ ∀𝑦𝑆 𝑥 𝑦))
15 breq1 4803 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 𝑤𝑥 𝑤))
1614, 15imbi12d 333 . . . . . . 7 (𝑧 = 𝑥 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑤) ↔ (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤)))
1716rspcv 3441 . . . . . 6 (𝑥𝐵 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤) → (∀𝑦𝑆 𝑥 𝑦𝑥 𝑤)))
1810, 11, 12, 17syl3c 66 . . . . 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 . . . . . . . 8 ((𝑤 𝑥𝑥 𝑤) ↔ (𝑥 𝑤𝑤 𝑥))
20 poslubmo.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
21 poslubmo.l . . . . . . . . 9 = (le‘𝐾)
2220, 21posasymb 17149 . . . . . . . 8 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑥 𝑤𝑤 𝑥) ↔ 𝑥 = 𝑤))
2319, 22syl5bb 272 . . . . . . 7 ((𝐾 ∈ Poset ∧ 𝑥𝐵𝑤𝐵) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
24233expb 1114 . . . . . 6 ((𝐾 ∈ Poset ∧ (𝑥𝐵𝑤𝐵)) → ((𝑤 𝑥𝑥 𝑤) ↔ 𝑥 = 𝑤))
2524ad4ant13 1207 . . . . 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 994 . . . 4 ((((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) ∧ ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))) → 𝑥 = 𝑤)
2726ex 449 . . 3 (((𝐾 ∈ Poset ∧ 𝑆𝐵) ∧ (𝑥𝐵𝑤𝐵)) → (((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ∧ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))) → 𝑥 = 𝑤))
2827ralrimivva 3105 . 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 4803 . . . . 5 (𝑥 = 𝑤 → (𝑥 𝑦𝑤 𝑦))
3029ralbidv 3120 . . . 4 (𝑥 = 𝑤 → (∀𝑦𝑆 𝑥 𝑦 ↔ ∀𝑦𝑆 𝑤 𝑦))
31 breq2 4804 . . . . . 6 (𝑥 = 𝑤 → (𝑧 𝑥𝑧 𝑤))
3231imbi2d 329 . . . . 5 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3332ralbidv 3120 . . . 4 (𝑥 = 𝑤 → (∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥) ↔ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤)))
3430, 33anbi12d 749 . . 3 (𝑥 = 𝑤 → ((∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦𝑆 𝑤 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑤))))
3534rmo4 3536 . 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 224 1 ((𝐾 ∈ Poset ∧ 𝑆𝐵) → ∃*𝑥𝐵 (∀𝑦𝑆 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑆 𝑧 𝑦𝑧 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wral 3046  ∃*wrmo 3049  wss 3711   class class class wbr 4800  cfv 6045  Basecbs 16055  lecple 16146  Posetcpo 17137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-nul 4937
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-br 4801  df-iota 6008  df-fv 6053  df-preset 17125  df-poset 17143
This theorem is referenced by: (None)
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