Theorem glbeldm2d 48756

Description: Member of the domain of the greatest lower bound function of a poset. (Contributed by Zhi Wang, 29-Sep-2024.)

Hypotheses

Ref	Expression
lubeldm2d.b	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lubeldm2d.l	⊢ (𝜑 → ≤ = (le‘𝐾))
glbeldm2d.g	⊢ (𝜑 → 𝐺 = (glb‘𝐾))
glbeldm2d.p	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
glbeldm2d.k	⊢ (𝜑 → 𝐾 ∈ Poset)

Assertion

Ref	Expression
glbeldm2d	⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

Distinct variable groups:   𝑥,𝐾,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐺(𝑥,𝑦,𝑧)   ≤ (𝑥,𝑦,𝑧)

Proof of Theorem glbeldm2d

Step	Hyp	Ref	Expression
1		eqid 2735	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2735	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2735	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		biid 261	. . 3 ⊢ ((∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
5		glbeldm2d.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Poset)
6	1, 2, 3, 4, 5	glbeldm2 48754	. 2 ⊢ (𝜑 → (𝑆 ∈ dom (glb‘𝐾) ↔ (𝑆 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))))
7		glbeldm2d.g	. . . 4 ⊢ (𝜑 → 𝐺 = (glb‘𝐾))
8	7	dmeqd 5919	. . 3 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
9	8	eleq2d 2825	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ dom (glb‘𝐾)))
10		lubeldm2d.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
11	10	sseq2d 4028	. . 3 ⊢ (𝜑 → (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝐾)))
12		glbeldm2d.p	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
13		lubeldm2d.l	. . . . . . . . . . 11 ⊢ (𝜑 → ≤ = (le‘𝐾))
14	13	breqd 5159	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐾)𝑦))
15	14	ralbidv 3176	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦))
16	13	breqd 5159	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ≤ 𝑦 ↔ 𝑧(le‘𝐾)𝑦))
17	16	ralbidv 3176	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦))
18	13	breqd 5159	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ≤ 𝑥 ↔ 𝑧(le‘𝐾)𝑥))
19	17, 18	imbi12d 344	. . . . . . . . . 10 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
20	10, 19	raleqbidv 3344	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
21	15, 20	anbi12d 632	. . . . . . . 8 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
22	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
23	12, 22	bitrd 279	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
24	23	pm5.32da 579	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))))
25	10	eleq2d 2825	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
26	25	anbi1d 631	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) ↔ (𝑥 ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))))
27	24, 26	bitrd 279	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝜓) ↔ (𝑥 ∈ (Base‘𝐾) ∧ (∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))))
28	27	rexbidv2 3173	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 𝜓 ↔ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
29	11, 28	anbi12d 632	. 2 ⊢ (𝜑 → ((𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓) ↔ (𝑆 ⊆ (Base‘𝐾) ∧ ∃𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))))
30	6, 9, 29	3bitr4d 311	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963   class class class wbr 5148  dom cdm 5689  ‘cfv 6563  Basecbs 17245  lecple 17305  Posetcpo 18365  glbcglb 18368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-proset 18352  df-poset 18371  df-glb 18405

This theorem is referenced by:  ipoglbdm  48779