Theorem glb0N 39817

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 2762	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2762	. . 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 4354	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	glbval 18399	. 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 39809	. . 3 ⊢ (𝐾 ∈ OP → 1 ∈ (Base‘𝐾))
11		ral0 4452	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦
12	11	a1bi 364	. . . . . 6 ⊢ (𝑧(le‘𝐾)𝑥 ↔ (∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
13	12	ralbii 3108	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑧(le‘𝐾)𝑥 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑧(le‘𝐾)𝑦 → 𝑧(le‘𝐾)𝑥))
14		ral0 4452	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑥(le‘𝐾)𝑦
15	14	biantrur 538	. . . . 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 484	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 1 ∈ (Base‘𝐾))
18		breq1 5103	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧(le‘𝐾)𝑥 ↔ 1 (le‘𝐾)𝑥))
19	18	rspcv 3577	. . . . . . 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 39816	. . . . . 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 39815	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 𝑧(le‘𝐾) 1 )
24	23	adantlr 725	. . . . . . . . 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 5104	. . . . . . . . 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 3160	. . . . 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	bitr3id 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 7382	. 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 2797	1 ⊢ (𝐾 ∈ OP → (𝐺‘∅) = 1 )

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ⊆ wss 3904  ∅c0 4285   class class class wbr 5100  ‘cfv 6521  ℩crio 7352  Basecbs 17245  lecple 17293  glbcglb 18342  1.cp1 18454  OPcops 39796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-ov 7399  df-proset 18326  df-poset 18345  df-lub 18376  df-glb 18377  df-p1 18456  df-oposet 39800

This theorem is referenced by:  pmapglb2N  40395  pmapglb2xN  40396