Theorem uzublem 45380

Description: A set of reals, indexed by upper integers, is bound if and only if any upper part is bound. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
uzublem.1	⊢ Ⅎ𝑗𝜑
uzublem.2	⊢ Ⅎ𝑗𝑋
uzublem.3	⊢ (𝜑 → 𝑀 ∈ ℤ)
uzublem.4	⊢ 𝑍 = (ℤ≥‘𝑀)
uzublem.5	⊢ (𝜑 → 𝑌 ∈ ℝ)
uzublem.6	⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < )
uzublem.7	⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)
uzublem.8	⊢ (𝜑 → 𝐾 ∈ 𝑍)
uzublem.9	⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
uzublem.10	⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)

Assertion

Ref	Expression
uzublem	⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)

Distinct variable groups:   𝑥,𝐵   𝑗,𝐾   𝑗,𝑀   𝑥,𝑋   𝑥,𝑍   𝑥,𝑗

Allowed substitution hints:   𝜑(𝑥,𝑗)   𝐵(𝑗)   𝐾(𝑥)   𝑀(𝑥)   𝑊(𝑥,𝑗)   𝑋(𝑗)   𝑌(𝑥,𝑗)   𝑍(𝑗)

Proof of Theorem uzublem

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uzublem.7	. . 3 ⊢ 𝑋 = if(𝑊 ≤ 𝑌, 𝑌, 𝑊)
2		uzublem.5	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ℝ)
3		uzublem.6	. . . . . 6 ⊢ 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < )
4	3	a1i 11	. . . . 5 ⊢ (𝜑 → 𝑊 = sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ))
5		uzublem.1	. . . . . 6 ⊢ Ⅎ𝑗𝜑
6		ltso 11339	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 14011	. . . . . 6 ⊢ (𝜑 → (𝑀...𝐾) ∈ Fin)
9		uzublem.3	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ)
10		uzublem.8	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ 𝑍)
11		uzublem.4	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ≥‘𝑀)
12	11	eluzelz2 45353	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑍 → 𝐾 ∈ ℤ)
13	10, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
14	9	zred 12720	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
15	14	leidd 11827	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
16	10, 11	eleqtrdi 2849	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
17		eluzle 12889	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
19	9, 13, 9, 15, 18	elfzd 13552	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝐾))
20	19	ne0d 4348	. . . . . 6 ⊢ (𝜑 → (𝑀...𝐾) ≠ ∅)
21		fzssuz 13602	. . . . . . . . 9 ⊢ (𝑀...𝐾) ⊆ (ℤ≥‘𝑀)
22	11	eqcomi 2744	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) = 𝑍
23	21, 22	sseqtri 4032	. . . . . . . 8 ⊢ (𝑀...𝐾) ⊆ 𝑍
24		id 22	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑀...𝐾) → 𝑗 ∈ (𝑀...𝐾))
25	23, 24	sselid 3993	. . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝐾) → 𝑗 ∈ 𝑍)
26		uzublem.9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ∈ ℝ)
27	25, 26	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝐾)) → 𝐵 ∈ ℝ)
28	5, 7, 8, 20, 27	fisupclrnmpt 45348	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
29	4, 28	eqeltrd 2839	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
30	2, 29	ifcld 4577	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
31	1, 30	eqeltrid 2843	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
32	26	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ∈ ℝ)
33	2	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
34	31	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑋 ∈ ℝ)
35		uzublem.10	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)
36	35	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)
37		eqid 2735	. . . . . . . 8 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
38	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℤ)
39	11	eluzelz2 45353	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
40	39	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑗 ∈ ℤ)
41		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
42	37, 38, 40, 41	eluzd 45359	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑗 ∈ (ℤ≥‘𝐾))
43		rspa 3246	. . . . . . 7 ⊢ ((∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌 ∧ 𝑗 ∈ (ℤ≥‘𝐾)) → 𝐵 ≤ 𝑌)
44	36, 42, 43	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ≤ 𝑌)
45		max2 13226	. . . . . . . . 9 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
46	29, 2, 45	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
47	46, 1	breqtrrdi 5190	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
48	47	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
49	32, 33, 34, 44, 48	letrd 11416	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ≤ 𝑋)
50		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → ¬ 𝐾 ≤ 𝑗)
51		uzssre 12898	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
52	11, 51	eqsstri 4030	. . . . . . . . . 10 ⊢ 𝑍 ⊆ ℝ
53	52	sseli 3991	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
54	53	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝑗 ∈ ℝ)
55	52, 10	sselid 3993	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
56	55	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℝ)
57	54, 56	ltnled 11406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → (𝑗 < 𝐾 ↔ ¬ 𝐾 ≤ 𝑗))
58	50, 57	mpbird 257	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝑗 < 𝐾)
59	26	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ∈ ℝ)
60	3, 29	eqeltrrid 2844	. . . . . . . . 9 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
61	3, 60	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
62	61	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ∈ ℝ)
63	2, 61	ifcld 4577	. . . . . . . . 9 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
64	1, 63	eqeltrid 2843	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
65	64	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑋 ∈ ℝ)
66		simpll 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝜑)
67	9	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑀 ∈ ℤ)
68	13	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐾 ∈ ℤ)
69	11	eleq2i 2831	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
70	69	biimpi 216	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
71	70	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (ℤ≥‘𝑀))
72		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 < 𝐾)
73	71, 68, 72	elfzod 45350	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀..^𝐾))
74		elfzouz 13700	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ≥‘𝑀))
75	74, 22	eleqtrdi 2849	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ 𝑍)
76	73, 75, 39	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ ℤ)
77		eluzle 12889	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
78	70, 77	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
79	78	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑀 ≤ 𝑗)
80	73, 75, 53	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ ℝ)
81	55	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐾 ∈ ℝ)
82	80, 81, 72	ltled 11407	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ≤ 𝐾)
83	67, 68, 76, 79, 82	elfzd 13552	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀...𝐾))
84	5, 27	ralrimia 3256	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝐾)𝐵 ∈ ℝ)
85		fimaxre3 12212	. . . . . . . . . . 11 ⊢ (((𝑀...𝐾) ∈ Fin ∧ ∀𝑗 ∈ (𝑀...𝐾)𝐵 ∈ ℝ) → ∃𝑦 ∈ ℝ ∀𝑗 ∈ (𝑀...𝐾)𝐵 ≤ 𝑦)
86	8, 84, 85	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑗 ∈ (𝑀...𝐾)𝐵 ≤ 𝑦)
87	5, 27, 86	suprubrnmpt 45198	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (𝑀...𝐾)) → 𝐵 ≤ sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ))
88	66, 83, 87	syl2anc 584	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ≤ sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ))
89	88, 3	breqtrrdi 5190	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ≤ 𝑊)
90		max1 13224	. . . . . . . . . 10 ⊢ ((𝑊 ∈ ℝ ∧ 𝑌 ∈ ℝ) → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
91	29, 2, 90	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ≤ if(𝑊 ≤ 𝑌, 𝑌, 𝑊))
92	91, 1	breqtrrdi 5190	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
93	92	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ≤ 𝑋)
94	59, 62, 65, 89, 93	letrd 11416	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ≤ 𝑋)
95	58, 94	syldan 591	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝐵 ≤ 𝑋)
96	49, 95	pm2.61dan 813	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ≤ 𝑋)
97	96	ex 412	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → 𝐵 ≤ 𝑋))
98	5, 97	ralrimi 3255	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋)
99		nfv 1912	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋
100		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑗𝑥
101		uzublem.2	. . . . 5 ⊢ Ⅎ𝑗𝑋
102	100, 101	nfeq 2917	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
103		breq2 5152	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑋))
104	102, 103	ralbid 3271	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋))
105	99, 104	rspce 3611	. 2 ⊢ ((𝑋 ∈ ℝ ∧ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋) → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)
106	31, 98, 105	syl2anc 584	1 ⊢ (𝜑 → ∃𝑥 ∈ ℝ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888  ∀wral 3059  ∃wrex 3068  ifcif 4531   class class class wbr 5148   ↦ cmpt 5231   Or wor 5596  ran crn 5690  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  supcsup 9478  ℝcr 11152   < clt 11293   ≤ cle 11294  ℤcz 12611  ℤ_≥cuz 12876  ...cfz 13544  ..^cfzo 13691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692

This theorem is referenced by:  uzub  45381