Theorem glb0N 36489

Description: The greatest lower bound of the empty set is the unit element. (Contributed by NM, 5-Dec-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
glb0.g	⊢ 𝐺 = (glb‘𝐾)
glb0.u	⊢ 1 = (1.‘𝐾)

Assertion

Ref	Expression
glb0N	⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Proof of Theorem glb0N

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2798	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2798	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		glb0.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		biid 264	. . 3 ⊢ ((∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
5		id 22	. . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP)
6		0ss 4304	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	glbval 17599	. 2 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))))
9		glb0.u	. . . 4 ⊢ 1 = (1.‘𝐾)
10	1, 9	op1cl 36481	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11		ral0 4414	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
12	11	a1bi 366	. . . . . 6 ⊢ (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
13	12	ralbii 3133	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
14		ral0 4414	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
15	14	biantrur 534	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥) ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
16	13, 15	bitri 278	. . . 4 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)))
17	10	adantr 484	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18		breq1 5033	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 1 (le‘𝐾)𝑥))
19	18	rspcv 3566	. . . . . . 7 ⊢ ( 1 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 → 1 (le‘𝐾)𝑥))
20	17, 19	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 → 1 (le‘𝐾)𝑥))
21	1, 2, 9	op1le 36488	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ( 1 (le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
22	20, 21	sylibd 242	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 → 𝑥 = 1 ))
23	1, 2, 9	ople1 36487	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
24	23	adantlr 714	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
25	24	ex 416	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾) 1 ))
26		breq2 5034	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 𝑧(le‘𝐾) 1 ))
27	26	biimprcd 253	. . . . . . . 8 ⊢ (𝑧(le‘𝐾) 1 → (𝑥 = 1 → 𝑧(le‘𝐾)𝑥))
28	25, 27	syl6 35	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 1 → 𝑧(le‘𝐾)𝑥)))
29	28	com23 86	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾)𝑥)))
30	29	ralrimdv 3153	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → ∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥))
31	22, 30	impbid 215	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
32	16, 31	bitr3id 288	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥)) ↔ 𝑥 = 1 ))
33	10, 32	riota5 7122	. 2 ⊢ (𝐾 ∈ OP → (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))) = 1 )
34	8, 33	eqtrd 2833	1 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  ∅c0 4243   class class class wbr 5030  ‘cfv 6324  ℩crio 7092  Basecbs 16475  lecple 16564  glbcglb 17545  1.cp1 17640  OPcops 36468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-proset 17530  df-poset 17548  df-lub 17576  df-glb 17577  df-p1 17642  df-oposet 36472

This theorem is referenced by:  pmapglb2N  37067  pmapglb2xN  37068