Theorem glbeldm2d 50036

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 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2761	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		eqid 2761	. . 3 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		biid 264	. . 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 50034	. 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 5887	. . 3 ⊢ (𝜑 → dom 𝐺 = dom (glb‘𝐾))
9	8	eleq2d 2847	. 2 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ 𝑆 ∈ dom (glb‘𝐾)))
10		lubeldm2d.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
11	10	sseq2d 3963	. . 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 5114	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ≤ 𝑦 ↔ 𝑥(le‘𝐾)𝑦))
15	14	ralbidv 3186	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑥(le‘𝐾)𝑦))
16	13	breqd 5114	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑧 ≤ 𝑦 ↔ 𝑧(le‘𝐾)𝑦))
17	16	ralbidv 3186	. . . . . . . . . . 11 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 ↔ ∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦))
18	13	breqd 5114	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ≤ 𝑥 ↔ 𝑧(le‘𝐾)𝑥))
19	17, 18	imbi12d 347	. . . . . . . . . 10 ⊢ (𝜑 → ((∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥) ↔ (∀𝑦 ∈ 𝑆 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
20	10, 19	raleqbidv 3335	. . . . . . . . 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 644	. . . . . . . 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 486	. . . . . . 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 282	. . . . . 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 590	. . . . 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 2847	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐾)))
26	25	anbi1d 643	. . . . 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 282	. . . 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 3183	. . 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 644	. 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 314	1 ⊢ (𝜑 → (𝑆 ∈ dom 𝐺 ↔ (𝑆 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐵 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899   class class class wbr 5103  dom cdm 5651  ‘cfv 6537  Basecbs 17380  lecple 17428  Posetcpo 18474  glbcglb 18477

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-proset 18461  df-poset 18480  df-glb 18512

This theorem is used by:  ipoglbdm  50067