Theorem uzublem 42977

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 11064	. . . . . . 7 ⊢ < Or ℝ
7	6	a1i 11	. . . . . 6 ⊢ (𝜑 → < Or ℝ)
8		fzfid 13702	. . . . . 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 42950	. . . . . . . . 9 ⊢ (𝐾 ∈ 𝑍 → 𝐾 ∈ ℤ)
13	10, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
14	9	zred 12435	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℝ)
15	14	leidd 11550	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
16	10, 11	eleqtrdi 2850	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
17		eluzle 12604	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝐾)
18	16, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ≤ 𝐾)
19	9, 13, 9, 15, 18	elfzd 13256	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝐾))
20	19	ne0d 4270	. . . . . 6 ⊢ (𝜑 → (𝑀...𝐾) ≠ ∅)
21		fzssuz 13306	. . . . . . . . 9 ⊢ (𝑀...𝐾) ⊆ (ℤ≥‘𝑀)
22	11	eqcomi 2748	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) = 𝑍
23	21, 22	sseqtri 3958	. . . . . . . 8 ⊢ (𝑀...𝐾) ⊆ 𝑍
24		id 22	. . . . . . . 8 ⊢ (𝑗 ∈ (𝑀...𝐾) → 𝑗 ∈ (𝑀...𝐾))
25	23, 24	sselid 3920	. . . . . . 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 42945	. . . . 5 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
29	4, 28	eqeltrd 2840	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ℝ)
30	2, 29	ifcld 4506	. . 3 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
31	1, 30	eqeltrid 2844	. 2 ⊢ (𝜑 → 𝑋 ∈ ℝ)
32	26	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ∈ ℝ)
33	2	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ∈ ℝ)
34	31	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑋 ∈ ℝ)
35		uzublem.10	. . . . . . . 8 ⊢ (𝜑 → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)
36	35	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → ∀𝑗 ∈ (ℤ≥‘𝐾)𝐵 ≤ 𝑌)
37		eqid 2739	. . . . . . . 8 ⊢ (ℤ≥‘𝐾) = (ℤ≥‘𝐾)
38	13	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℤ)
39	11	eluzelz2 42950	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℤ)
40	39	ad2antlr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑗 ∈ ℤ)
41		simpr 485	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐾 ≤ 𝑗)
42	37, 38, 40, 41	eluzd 42956	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑗 ∈ (ℤ≥‘𝐾))
43		rspa 3133	. . . . . . 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 12930	. . . . . . . . 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 5117	. . . . . . 7 ⊢ (𝜑 → 𝑌 ≤ 𝑋)
48	47	ad2antrr 723	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝑌 ≤ 𝑋)
49	32, 33, 34, 44, 48	letrd 11141	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝐾 ≤ 𝑗) → 𝐵 ≤ 𝑋)
50		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → ¬ 𝐾 ≤ 𝑗)
51		uzssre 12613	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
52	11, 51	eqsstri 3956	. . . . . . . . . 10 ⊢ 𝑍 ⊆ ℝ
53	52	sseli 3918	. . . . . . . . 9 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ ℝ)
54	53	ad2antlr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝑗 ∈ ℝ)
55	52, 10	sselid 3920	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℝ)
56	55	ad2antrr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝐾 ∈ ℝ)
57	54, 56	ltnled 11131	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → (𝑗 < 𝐾 ↔ ¬ 𝐾 ≤ 𝑗))
58	50, 57	mpbird 256	. . . . . 6 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ ¬ 𝐾 ≤ 𝑗) → 𝑗 < 𝐾)
59	26	adantr 481	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ∈ ℝ)
60	3, 29	eqeltrrid 2845	. . . . . . . . 9 ⊢ (𝜑 → sup(ran (𝑗 ∈ (𝑀...𝐾) ↦ 𝐵), ℝ, < ) ∈ ℝ)
61	3, 60	eqeltrid 2844	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ ℝ)
62	61	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ∈ ℝ)
63	2, 61	ifcld 4506	. . . . . . . . 9 ⊢ (𝜑 → if(𝑊 ≤ 𝑌, 𝑌, 𝑊) ∈ ℝ)
64	1, 63	eqeltrid 2844	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ℝ)
65	64	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑋 ∈ ℝ)
66		simpll 764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝜑)
67	9	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑀 ∈ ℤ)
68	13	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐾 ∈ ℤ)
69	11	eleq2i 2831	. . . . . . . . . . . . . 14 ⊢ (𝑗 ∈ 𝑍 ↔ 𝑗 ∈ (ℤ≥‘𝑀))
70	69	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑗 ∈ 𝑍 → 𝑗 ∈ (ℤ≥‘𝑀))
71	70	ad2antlr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (ℤ≥‘𝑀))
72		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 < 𝐾)
73	71, 68, 72	elfzod 42947	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀..^𝐾))
74		elfzouz 13400	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ (ℤ≥‘𝑀))
75	74, 22	eleqtrdi 2850	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (𝑀..^𝐾) → 𝑗 ∈ 𝑍)
76	73, 75, 39	3syl 18	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ ℤ)
77		eluzle 12604	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑗)
78	70, 77	syl 17	. . . . . . . . . . 11 ⊢ (𝑗 ∈ 𝑍 → 𝑀 ≤ 𝑗)
79	78	ad2antlr 724	. . . . . . . . . 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 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐾 ∈ ℝ)
82	80, 81, 72	ltled 11132	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ≤ 𝐾)
83	67, 68, 76, 79, 82	elfzd 13256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑗 ∈ (𝑀...𝐾))
84	5, 27	ralrimia 3431	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑗 ∈ (𝑀...𝐾)𝐵 ∈ ℝ)
85		fimaxre3 11930	. . . . . . . . . . 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 42806	. . . . . . . . 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 5117	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝐵 ≤ 𝑊)
90		max1 12928	. . . . . . . . . 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 5117	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ≤ 𝑋)
93	92	ad2antrr 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑗 < 𝐾) → 𝑊 ≤ 𝑋)
94	59, 62, 65, 89, 93	letrd 11141	. . . . . 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 810	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝐵 ≤ 𝑋)
97	96	ex 413	. . 3 ⊢ (𝜑 → (𝑗 ∈ 𝑍 → 𝐵 ≤ 𝑋))
98	5, 97	ralrimi 3142	. 2 ⊢ (𝜑 → ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋)
99		nfv 1918	. . 3 ⊢ Ⅎ𝑥∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋
100		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑗𝑥
101		uzublem.2	. . . . 5 ⊢ Ⅎ𝑗𝑋
102	100, 101	nfeq 2921	. . . 4 ⊢ Ⅎ𝑗 𝑥 = 𝑋
103		breq2 5079	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐵 ≤ 𝑥 ↔ 𝐵 ≤ 𝑋))
104	102, 103	ralbid 3162	. . 3 ⊢ (𝑥 = 𝑋 → (∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑥 ↔ ∀𝑗 ∈ 𝑍 𝐵 ≤ 𝑋))
105	99, 104	rspce 3551	. 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 396   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2107  Ⅎwnfc 2888  ∀wral 3065  ∃wrex 3066  ifcif 4460   class class class wbr 5075   ↦ cmpt 5158   Or wor 5503  ran crn 5591  ‘cfv 6437  (class class class)co 7284  Fincfn 8742  supcsup 9208  ℝcr 10879   < clt 11018   ≤ cle 11019  ℤcz 12328  ℤ_≥cuz 12591  ...cfz 13248  ..^cfzo 13391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-nn 11983  df-n0 12243  df-z 12329  df-uz 12592  df-fz 13249  df-fzo 13392

This theorem is referenced by:  uzub  42978