Theorem glb0N 39392

Description: The greatest lower bound of the empty set is the unity 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 2734	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2734	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		glb0.g	. . 3 ⊢ 𝐺 = (glb‘𝐾)
4		biid 261	. . 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 18288	. 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 39384	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11		ral0 4449	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
12	11	a1bi 362	. . . . . 6 ⊢ (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
13	12	ralbii 3080	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
14		ral0 4449	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
15	14	biantrur 530	. . . . 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 275	. . . 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 480	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18		breq1 5099	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 1 (le‘𝐾)𝑥))
19	18	rspcv 3570	. . . . . . 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 39391	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ( 1 (le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
22	20, 21	sylibd 239	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 → 𝑥 = 1 ))
23	1, 2, 9	ople1 39390	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
24	23	adantlr 715	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
25	24	ex 412	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 𝑧(le‘𝐾) 1 ))
26		breq2 5100	. . . . . . . . 9 ⊢ (𝑥 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 𝑧(le‘𝐾) 1 ))
27	26	biimprcd 250	. . . . . . . 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 3132	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 1 → ∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥))
31	22, 30	impbid 212	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ 𝑥 = 1 ))
32	16, 31	bitr3id 285	. . 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 7342	. 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 2769	1 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ⊆ wss 3899  ∅c0 4283   class class class wbr 5096  ‘cfv 6490  ℩crio 7312  Basecbs 17134  lecple 17182  glbcglb 18231  1.cp1 18343  OPcops 39371

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-ov 7359  df-proset 18215  df-poset 18234  df-lub 18265  df-glb 18266  df-p1 18345  df-oposet 39375

This theorem is referenced by:  pmapglb2N  39970  pmapglb2xN  39971