Theorem uzublem 45429

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 11196	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13880	. . . . . 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 45402	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑍 → 𝐾 ∈ ℤ)
13	10, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
14	9	zred 12580	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
15	14	leidd 11686	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
16	10, 11	eleqtrdi 2838	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
17		eluzle 12748	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
19	9, 13, 9, 15, 18	elfzd 13418	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝐾))
20	19	ne0d 4293	. . . . . 6 ⊢ (𝜑 → (𝑀...𝐾) ≠ ∅)
21		fzssuz 13468	. . . . . . . . 9 ⊢ (𝑀...𝐾) ⊆ (ℤ≥‘𝑀)
22	11	eqcomi 2738	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) = 𝑍
23	21, 22	sseqtri 3984	. . . . . . . 8 ⊢ (𝑀...𝐾) ⊆ 𝑍
24		id 22	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑀...𝐾) → 𝑗 ∈ (𝑀...𝐾))
25	23, 24	sselid 3933	. . . . . . 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 45397	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
29	4, 28	eqeltrd 2828	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
30	2, 29	ifcld 4523	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
31	1, 30	eqeltrid 2832	. 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 2729	. . . . . . . 8 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
38	13	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℤ)
39	11	eluzelz2 45402	. . . . . . . . 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 45408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑗 ∈ (ℤ≥‘𝐾))
43		rspa 3218	. . . . . . 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 13089	. . . . . . . . 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 5134	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
48	47	ad2antrr 726	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
49	32, 33, 34, 44, 48	letrd 11273	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ≤ 𝑋)
50		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → ¬ 𝐾 ≤ 𝑗)
51		uzssre 12757	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
52	11, 51	eqsstri 3982	. . . . . . . . . 10 ⊢ 𝑍 ⊆ ℝ
53	52	sseli 3931	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
54	53	ad2antlr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝑗 ∈ ℝ)
55	52, 10	sselid 3933	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
56	55	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℝ)
57	54, 56	ltnled 11263	. . . . . . 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 2833	. . . . . . . . 9 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
61	3, 60	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
62	61	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ∈ ℝ)
63	2, 61	ifcld 4523	. . . . . . . . 9 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
64	1, 63	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
65	64	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑋 ∈ ℝ)
66		simpll 766	. . . . . . . . 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 2820	. . . . . . . . . . . . . 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 45399	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀..^𝐾))
74		elfzouz 13566	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ≥‘𝑀))
75	74, 22	eleqtrdi 2838	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ 𝑍)
76	73, 75, 39	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ ℤ)
77		eluzle 12748	. . . . . . . . . . . 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 11264	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ≤ 𝐾)
83	67, 68, 76, 79, 82	elfzd 13418	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀...𝐾))
84	5, 27	ralrimia 3228	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝐾)𝐵 ∈ ℝ)
85		fimaxre3 12071	. . . . . . . . . . 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 45251	. . . . . . . . 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 5134	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ≤ 𝑊)
90		max1 13087	. . . . . . . . . 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 5134	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
93	92	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ≤ 𝑋)
94	59, 62, 65, 89, 93	letrd 11273	. . . . . 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 812	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ≤ 𝑋)
97	96	ex 412	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → 𝐵 ≤ 𝑋))
98	5, 97	ralrimi 3227	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋)
99		nfv 1914	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋
100		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑗𝑥
101		uzublem.2	. . . . 5 ⊢ Ⅎ𝑗𝑋
102	100, 101	nfeq 2905	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
103		breq2 5096	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑋))
104	102, 103	ralbid 3242	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋))
105	99, 104	rspce 3566	. 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 1540  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2876  ∀wral 3044  ∃wrex 3053  ifcif 4476   class class class wbr 5092   ↦ cmpt 5173   Or wor 5526  ran crn 5620  ‘cfv 6482  (class class class)co 7349  Fincfn 8872  supcsup 9330  ℝcr 11008   < clt 11149   ≤ cle 11150  ℤcz 12471  ℤ_≥cuz 12735  ...cfz 13410  ..^cfzo 13557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086  ax-pre-sup 11087

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-sup 9332  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558

This theorem is referenced by:  uzub  45430