Theorem exfzdc 10586

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 9867	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
4	1, 2, 3	syl2anc 411	. . . 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 10366	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
8		oveq2 6058	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
9	8	rexeqdv 2748	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
10	9	dcbid 846	. . . . . 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 6058	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
13	12	rexeqdv 2748	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
14	13	dcbid 846	. . . . . 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 6058	. . . . . . . 8 ⊢ (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
17	16	rexeqdv 2748	. . . . . . 7 ⊢ (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
18	17	dcbid 846	. . . . . 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 6058	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
21	20	rexeqdv 2748	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
22	21	dcbid 846	. . . . . 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 10365	. . . . . . . . 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 2615	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
28	27	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29		nfsbc1v 3061	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑀 / 𝑛]𝜓
30	29	nfdc 1707	. . . . . . . . 9 ⊢ Ⅎ𝑛DECID [𝑀 / 𝑛]𝜓
31		sbceq1a 3052	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ [𝑀 / 𝑛]𝜓))
32	31	dcbid 846	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (DECID 𝜓 ↔ DECID [𝑀 / 𝑛]𝜓))
33	30, 32	rspc 2915	. . . . . . . 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 10400	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑀) = {𝑀})
38	37	rexeqdv 2748	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39		rexsns 3728	. . . . . . . . 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 846	. . . . . . 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 10572	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
46	45	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑦 + 1) ∈ (𝑀...𝑁))
47	27	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
48		nfsbc1v 3061	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛[(𝑦 + 1) / 𝑛]𝜓
49	48	nfdc 1707	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50		sbceq1a 3052	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓))
51	50	dcbid 846	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑦 + 1) → (DECID 𝜓 ↔ DECID [(𝑦 + 1) / 𝑛]𝜓))
52	49, 51	rspc 2915	. . . . . . . . . . . 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 3728	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ {(𝑦 + 1)}𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓)
55	54	dcbii 848	. . . . . . . . . . 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 944	. . . . . . . . . 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 3399	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
60	59	dcbii 848	. . . . . . . . 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 10485	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ≥‘𝑀))
63	62	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ≥‘𝑀))
64		fzsuc 10403	. . . . . . . . . . 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 2748	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
67	66	dcbid 846	. . . . . . . 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 10585	. . . 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 3526	. . . . 5 ⊢ ¬ ∃𝑛 ∈ ∅ 𝜓
75		zltnle 9623	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
76	2, 1, 75	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
77	76	biimpar 297	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → 𝑁 < 𝑀)
78		fzn 10376	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
79	1, 2, 78	syl2anc 411	. . . . . . . 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 2748	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
83	74, 82	mtbiri 682	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
84	83	olcd 742	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85		df-dc 843	. . 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 9654	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
88		exmiddc 844	. . . 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 411	. 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
91	73, 86, 90	mpjaodan 806	1 ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  [wsbc 3042   ∪ cun 3209  ∅c0 3508  {csn 3689   class class class wbr 4109  ‘cfv 5352  (class class class)co 6050  1c1 8128   + caddc 8130   < clt 8308   ≤ cle 8309  ℤcz 9577  ℤ_≥cuz 9853  ...cfz 10342  ..^cfzo 10476

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477

This theorem is referenced by:  prmind2  12817  4sqlemafi  13093  4sqexercise1  13096  4sqexercise2  13097  4sqlemsdc  13098