Theorem exfzdc 10637

Description: Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)

Hypotheses

Ref	Expression
exfzdc.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
exfzdc.2	⊢ (𝜑 → 𝑁 ∈ ℤ)
exfzdc.3	⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)

Assertion

Ref	Expression
exfzdc	⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

Distinct variable groups:   𝑛,𝑀   𝑛,𝑁   𝜑,𝑛

Allowed substitution hint:   𝜓(𝑛)

Proof of Theorem exfzdc

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exfzdc.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		exfzdc.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
3		eluz 9914	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
4	1, 2, 3	syl2anc 415	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
5	4	biimpar 297	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (ℤ≥‘𝑀))
6		simpl 109	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝜑)
7		eluzfz2 10415	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
8		oveq2 6083	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
9	8	rexeqdv 2756	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
10	9	dcbid 850	. . . . . 6 ⊢ (𝑤 = 𝑀 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓))
11	10	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑀 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓)))
12		oveq2 6083	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
13	12	rexeqdv 2756	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
14	13	dcbid 850	. . . . . 6 ⊢ (𝑤 = 𝑦 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓))
15	14	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓)))
16		oveq2 6083	. . . . . . . 8 ⊢ (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
17	16	rexeqdv 2756	. . . . . . 7 ⊢ (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
18	17	dcbid 850	. . . . . 6 ⊢ (𝑤 = (𝑦 + 1) → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
19	18	imbi2d 230	. . . . 5 ⊢ (𝑤 = (𝑦 + 1) → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
20		oveq2 6083	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
21	20	rexeqdv 2756	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
22	21	dcbid 850	. . . . . 6 ⊢ (𝑤 = 𝑁 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
23	22	imbi2d 230	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)))
24		eluzfz1 10414	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	24	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (𝑀...𝑁))
26		exfzdc.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
27	26	ralrimiva 2623	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29		nfsbc1v 3070	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑀 / 𝑛]𝜓
30	29	nfdc 1711	. . . . . . . . 9 ⊢ Ⅎ𝑛DECID [𝑀 / 𝑛]𝜓
31		sbceq1a 3061	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ [𝑀 / 𝑛]𝜓))
32	31	dcbid 850	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (DECID 𝜓 ↔ DECID [𝑀 / 𝑛]𝜓))
33	30, 32	rspc 2923	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓 → DECID [𝑀 / 𝑛]𝜓))
34	25, 28, 33	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID [𝑀 / 𝑛]𝜓)
35	1	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ)
36		fzsn 10450	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑀) = {𝑀})
38	37	rexeqdv 2756	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39		rexsns 3744	. . . . . . . . 9 ⊢ (∃𝑛 ∈ {𝑀}𝜓 ↔ [𝑀 / 𝑛]𝜓)
40	38, 39	bitrdi 196	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ [𝑀 / 𝑛]𝜓))
41	40	dcbid 850	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ DECID [𝑀 / 𝑛]𝜓))
42	34, 41	mpbird 167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓)
43	42	expcom 116	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓))
44		simpr 110	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓)
45		fzofzp1 10623	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
46	45	ad2antrr 492	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑦 + 1) ∈ (𝑀...𝑁))
47	27	ad2antlr 493	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
48		nfsbc1v 3070	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛[(𝑦 + 1) / 𝑛]𝜓
49	48	nfdc 1711	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50		sbceq1a 3061	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓))
51	50	dcbid 850	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑦 + 1) → (DECID 𝜓 ↔ DECID [(𝑦 + 1) / 𝑛]𝜓))
52	49, 51	rspc 2923	. . . . . . . . . . . 12 ⊢ ((𝑦 + 1) ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓 → DECID [(𝑦 + 1) / 𝑛]𝜓))
53	46, 47, 52	sylc 62	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID [(𝑦 + 1) / 𝑛]𝜓)
54		rexsns 3744	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ {(𝑦 + 1)}𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓)
55	54	dcbii 852	. . . . . . . . . . 11 ⊢ (DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓 ↔ DECID [(𝑦 + 1) / 𝑛]𝜓)
56	53, 55	sylibr 134	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓)
57		dcor 948	. . . . . . . . . 10 ⊢ (DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓 → (DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓 → DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓)))
58	44, 56, 57	sylc 62	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
59		rexun 3409	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
60	59	dcbii 852	. . . . . . . . 9 ⊢ (DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
61	58, 60	sylibr 134	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓)
62		elfzouz 10536	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ≥‘𝑀))
63	62	ad2antrr 492	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ≥‘𝑀))
64		fzsuc 10453	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
65	63, 64	syl 14	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
66	65	rexeqdv 2756	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
67	66	dcbid 850	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
68	61, 67	mpbird 167	. . . . . . 7 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)
69	68	exp31 364	. . . . . 6 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜑 → (DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
70	69	a2d 26	. . . . 5 ⊢ (𝑦 ∈ (𝑀..^𝑁) → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
71	11, 15, 19, 23, 43, 70	fzind2 10636	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
72	7, 71	syl 14	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
73	5, 6, 72	sylc 62	. 2 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)
74		rex0 3539	. . . . 5 ⊢ ¬ ∃𝑛 ∈ ∅ 𝜓
75		zltnle 9669	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
76	2, 1, 75	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
77	76	biimpar 297	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → 𝑁 < 𝑀)
78		fzn 10425	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
79	1, 2, 78	syl2anc 415	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
80	79	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
81	77, 80	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (𝑀...𝑁) = ∅)
82	81	rexeqdv 2756	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
83	74, 82	mtbiri 686	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
84	83	olcd 746	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85		df-dc 847	. . 3 ⊢ (DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
86	84, 85	sylibr 134	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)
87		zdcle 9700	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
88		exmiddc 848	. . . 4 ⊢ (DECID 𝑀 ≤ 𝑁 → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
89	87, 88	syl 14	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
90	1, 2, 89	syl2anc 415	. 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
91	73, 86, 90	mpjaodan 810	1 ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  [wsbc 3051   ∪ cun 3218  ∅c0 3520  {csn 3705   class class class wbr 4125  ‘cfv 5372  (class class class)co 6075  1c1 8170   + caddc 8172   < clt 8350   ≤ cle 8351  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  ..^cfzo 10527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528

This theorem is referenced by:  prmind2  12876  4sqlemafi  13152  4sqexercise1  13155  4sqexercise2  13156  4sqlemsdc  13157