Theorem lubprop 18556

Description: Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
lubprop.b	⊢ 𝐵 = (Base‘𝐾)
lubprop.l	⊢ ≤ = (le‘𝐾)
lubprop.u	⊢ 𝑈 = (lub‘𝐾)
lubprop.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lubprop.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)

Assertion

Ref	Expression
lubprop	⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))

Distinct variable groups:   𝑧,𝐵   𝑦,𝑧,𝐾   𝑦,𝑆,𝑧   𝑦, ≤   𝑦,𝑈,𝑧

Allowed substitution hints:   𝜑(𝑦, 𝑧)   𝐵(𝑦)   ≤ (𝑧)   𝑉(𝑦, 𝑧)

Proof of Theorem lubprop

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lubprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		lubprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		lubprop.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
4		biid 264	. . . 4 ⊢ ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
5		lubprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		lubprop.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	1, 2, 3, 5, 6	lubelss 18552	. . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
8	1, 2, 3, 4, 5, 7	lubval 18554	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) = (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
9	8	eqcomd 2769	. 2 ⊢ (𝜑 → (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (𝑈‘𝑆))
10	1, 3, 5, 6	lubcl 18555	. . 3 ⊢ (𝜑 → (𝑈‘𝑆) ∈ 𝐵)
11	1, 2, 3, 4, 5, 6	lubeu 18553	. . 3 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
12		breq2 5110	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑦 ≤ 𝑥 ↔ 𝑦 ≤ (𝑈‘𝑆)))
13	12	ralbidv 3189	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆)))
14		breq1 5109	. . . . . . 7 ⊢ (𝑥 = (𝑈‘𝑆) → (𝑥 ≤ 𝑧 ↔ (𝑈‘𝑆) ≤ 𝑧))
15	14	imbi2d 343	. . . . . 6 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
16	15	ralbidv 3189	. . . . 5 ⊢ (𝑥 = (𝑈‘𝑆) → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧) ↔ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
17	13, 16	anbi12d 644	. . . 4 ⊢ (𝑥 = (𝑈‘𝑆) → ((∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧))))
18	17	riota2 7422	. . 3 ⊢ (((𝑈‘𝑆) ∈ 𝐵 ∧ ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) → ((∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)) ↔ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (𝑈‘𝑆)))
19	10, 11, 18	syl2anc 596	. 2 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)) ↔ (℩𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))) = (𝑈‘𝑆)))
20	9, 19	mpbird 260	1 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃!wreu 3367   class class class wbr 5106  dom cdm 5655  ‘cfv 6555  ℩crio 7396  Basecbs 17413  lecple 17461  lubclub 18509

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-lub 18544

This theorem is used by:  luble  18557  lublem  18710