Theorem glb0N 36344

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 2821	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2821	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		glb0.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		biid 263	. . 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 4350	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	glbval 17607	. 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 36336	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11		ral0 4456	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
12	11	a1bi 365	. . . . . 6 ⊢ (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
13	12	ralbii 3165	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
14		ral0 4456	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
15	14	biantrur 533	. . . . 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 277	. . . 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 483	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18		breq1 5069	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 1 (le‘𝐾)𝑥))
19	18	rspcv 3618	. . . . . . 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 36343	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ( 1 (le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
22	20, 21	sylibd 241	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 → 𝑥 = 1 ))
23	1, 2, 9	ople1 36342	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
24	23	adantlr 713	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
25	24	ex 415	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾) 1 ))
26		breq2 5070	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 𝑧(le‘𝐾) 1 ))
27	26	biimprcd 252	. . . . . . . 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 3188	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → ∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥))
31	22, 30	impbid 214	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
32	16, 31	syl5bbr 287	. . 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 7143	. 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 2856	1 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ⊆ wss 3936  ∅c0 4291   class class class wbr 5066  ‘cfv 6355  ℩crio 7113  Basecbs 16483  lecple 16572  glbcglb 17553  1.cp1 17648  OPcops 36323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-proset 17538  df-poset 17556  df-lub 17584  df-glb 17585  df-p1 17650  df-oposet 36327

This theorem is referenced by:  pmapglb2N  36922  pmapglb2xN  36923