Theorem lub0N 35882

Description: The least upper bound of the empty set is the zero element. (Contributed by NM, 15-Sep-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
lub0.u	⊢ 1 = (lub‘𝐾)
lub0.z	⊢ 0 = (0.‘𝐾)

Assertion

Ref	Expression
lub0N	⊢ (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Proof of Theorem lub0N

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

Step	Hyp	Ref	Expression
1		eqid 2795	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2795	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		lub0.u	. . 3 ⊢ 1 = (lub‘𝐾)
4		biid 262	. . 3 ⊢ ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
5		id 22	. . 3 ⊢ (𝐾 ∈ OP → 𝐾 ∈ OP)
6		0ss 4274	. . . 4 ⊢ ∅ ⊆ (Base‘𝐾)
7	6	a1i 11	. . 3 ⊢ (𝐾 ∈ OP → ∅ ⊆ (Base‘𝐾))
8	1, 2, 3, 4, 5, 7	lubval 17428	. 2 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))))
9		lub0.z	. . . 4 ⊢ 0 = (0.‘𝐾)
10	1, 9	op0cl 35877	. . 3 ⊢ (𝐾 ∈ OP → 0 ∈ (Base‘𝐾))
11		ral0 4374	. . . . . . 7 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧
12	11	a1bi 364	. . . . . 6 ⊢ (𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
13	12	ralbii 3132	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))
14		ral0 4374	. . . . . 6 ⊢ ∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥
15	14	biantrur 531	. . . . 5 ⊢ (∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧) ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
16	13, 15	bitri 276	. . . 4 ⊢ (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ (∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)))
17	10	adantr 481	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → 0 ∈ (Base‘𝐾))
18		breq2 4970	. . . . . . . 8 ⊢ (𝑧 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 𝑥(le‘𝐾) 0 ))
19	18	rspcv 3555	. . . . . . 7 ⊢ ( 0 ∈ (Base‘𝐾) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥(le‘𝐾) 0 ))
20	17, 19	syl 17	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥(le‘𝐾) 0 ))
21	1, 2, 9	ople0 35880	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥(le‘𝐾) 0 ↔ 𝑥 = 0 ))
22	20, 21	sylibd 240	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 → 𝑥 = 0 ))
23	1, 2, 9	op0le 35879	. . . . . . . . . 10 ⊢ ((𝐾 ∈ OP ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
24	23	adantlr 711	. . . . . . . . 9 ⊢ (((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) ∧ 𝑧 ∈ (Base‘𝐾)) → 0 (le‘𝐾)𝑧)
25	24	ex 413	. . . . . . . 8 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → 0 (le‘𝐾)𝑧))
26		breq1 4969	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥(le‘𝐾)𝑧 ↔ 0 (le‘𝐾)𝑧))
27	26	biimprcd 251	. . . . . . . 8 ⊢ ( 0 (le‘𝐾)𝑧 → (𝑥 = 0 → 𝑥(le‘𝐾)𝑧))
28	25, 27	syl6 35	. . . . . . 7 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑧 ∈ (Base‘𝐾) → (𝑥 = 0 → 𝑥(le‘𝐾)𝑧)))
29	28	com23 86	. . . . . 6 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → (𝑧 ∈ (Base‘𝐾) → 𝑥(le‘𝐾)𝑧)))
30	29	ralrimdv 3155	. . . . 5 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (𝑥 = 0 → ∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧))
31	22, 30	impbid 213	. . . 4 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → (∀𝑧 ∈ (Base‘𝐾)𝑥(le‘𝐾)𝑧 ↔ 𝑥 = 0 ))
32	16, 31	syl5bbr 286	. . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑥 ∈ (Base‘𝐾)) → ((∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧)) ↔ 𝑥 = 0 ))
33	10, 32	riota5 7008	. 2 ⊢ (𝐾 ∈ OP → (℩𝑥 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑥 ∧ ∀𝑧 ∈ (Base‘𝐾)(∀𝑦 ∈ ∅ 𝑦(le‘𝐾)𝑧 → 𝑥(le‘𝐾)𝑧))) = 0 )
34	8, 33	eqtrd 2831	1 ⊢ (𝐾 ∈ OP → ( 1 ‘∅) = 0 )

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105   ⊆ wss 3863  ∅c0 4215   class class class wbr 4966  ‘cfv 6230  ℩crio 6981  Basecbs 16317  lecple 16406  lubclub 17386  0.cp0 17481  OPcops 35865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5086  ax-sep 5099  ax-nul 5106  ax-pow 5162  ax-pr 5226

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3710  df-csb 3816  df-dif 3866  df-un 3868  df-in 3870  df-ss 3878  df-nul 4216  df-if 4386  df-pw 4459  df-sn 4477  df-pr 4479  df-op 4483  df-uni 4750  df-iun 4831  df-br 4967  df-opab 5029  df-mpt 5046  df-id 5353  df-xp 5454  df-rel 5455  df-cnv 5456  df-co 5457  df-dm 5458  df-rn 5459  df-res 5460  df-ima 5461  df-iota 6194  df-fun 6232  df-fn 6233  df-f 6234  df-f1 6235  df-fo 6236  df-f1o 6237  df-fv 6238  df-riota 6982  df-ov 7024  df-proset 17372  df-poset 17390  df-lub 17418  df-glb 17419  df-p0 17483  df-oposet 35869

This theorem is referenced by: (None)