Theorem seq3z 10602

Description: If the operation + has an absorbing element 𝑍 (a.k.a. zero element), then any sequence containing a 𝑍 evaluates to 𝑍. (Contributed by Mario Carneiro, 27-May-2014.) (Revised by Jim Kingdon, 23-Apr-2023.)

Hypotheses

Ref	Expression
seq3homo.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
seq3homo.2	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seqz.3	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍)
seqz.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍)
seqz.5	⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
seqz.7	⊢ (𝜑 → (𝐹‘𝐾) = 𝑍)

Assertion

Ref	Expression
seq3z	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Distinct variable groups:   𝑥,𝑦,𝐹   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝜑,𝑥,𝑦   𝑥,𝐾,𝑦   𝑥, + ,𝑦   𝑥,𝑆,𝑦   𝑥,𝑍,𝑦

Proof of Theorem seq3z

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

Step	Hyp	Ref	Expression
1		seqz.5	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
2		elfzuz3 10091	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
4		fveqeq2 5564	. . . 4 ⊢ (𝑤 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
5	4	imbi2d 230	. . 3 ⊢ (𝑤 = 𝐾 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)))
6		fveqeq2 5564	. . . 4 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
7	6	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
8		fveqeq2 5564	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
9	8	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
10		fveqeq2 5564	. . . 4 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
11	10	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
12		elfzuz 10090	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
13	1, 12	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
14		eluzelz 9604	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
15	13, 14	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝑥 ∈ (ℤ≥‘𝐾))
17	13	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀))
18		uztrn 9612	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ (ℤ≥‘𝑀))
19	16, 17, 18	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝑥 ∈ (ℤ≥‘𝑀))
20		seq3homo.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
21	19, 20	syldan 282	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → (𝐹‘𝑥) ∈ 𝑆)
22		seq3homo.1	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
23	15, 21, 22	seq3-1 10536	. . . . . . 7 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾))
24		seqz.7	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍)
25	23, 24	eqtrd 2226	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍)
26		seqeq1 10524	. . . . . . . 8 ⊢ (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹))
27	26	fveq1d 5557	. . . . . . 7 ⊢ (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
28	27	eqeq1d 2202	. . . . . 6 ⊢ (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
29	25, 28	syl5ibcom 155	. . . . 5 ⊢ (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
30		eluzel2 9600	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
31	13, 30	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
32	31	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
33		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))
34	20	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
35	22	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	32, 33, 34, 35	seq3m1 10547	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)))
37	24	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝐾) = 𝑍)
38	37	oveq2d 5935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
39		oveq1 5926	. . . . . . . . 9 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
40	39	eqeq1d 2202	. . . . . . . 8 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → ((𝑥 + 𝑍) = 𝑍 ↔ ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍))
41		seqz.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑍)
42	41	ralrimiva 2567	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍)
43	42	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍)
44		eqid 2193	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
45	44, 32, 34, 35	seqf 10538	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
46		eluzp1m1 9619	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ≥‘𝑀))
47	31, 46	sylan 283	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ≥‘𝑀))
48	45, 47	ffvelcdmd 5695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆)
49	40, 43, 48	rspcdva 2870	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)
50	36, 38, 49	3eqtrd 2230	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
51	50	ex 115	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
52		uzp1 9629	. . . . . 6 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))))
53	13, 52	syl 14	. . . . 5 ⊢ (𝜑 → (𝐾 = 𝑀 ∨ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))))
54	29, 51, 53	mpjaod 719	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
55	54	a1i 9	. . 3 ⊢ (𝐾 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
56		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝑘 ∈ (ℤ≥‘𝐾))
57	13	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀))
58		uztrn 9612	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
59	56, 57, 58	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝑘 ∈ (ℤ≥‘𝑀))
60	20	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
61	22	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
62	59, 60, 61	seq3p1 10539	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
63	62	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
64		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
65	64	oveq1d 5934	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
66		oveq2 5927	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
67	66	eqeq1d 2202	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘(𝑘 + 1)) → ((𝑍 + 𝑥) = 𝑍 ↔ (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍))
68		seqz.3	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑍 + 𝑥) = 𝑍)
69	68	ralrimiva 2567	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍)
70	69	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍)
71		fveq2 5555	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
72	71	eleq1d 2262	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
73	20	ralrimiva 2567	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
74	73	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
75		peano2uz 9651	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
76	59, 75	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝑘 + 1) ∈ (ℤ≥‘𝑀))
77	72, 74, 76	rspcdva 2870	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
78	67, 70, 77	rspcdva 2870	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
79	78	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
80	63, 65, 79	3eqtrd 2230	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
81	80	ex 115	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
82	81	expcom 116	. . . 4 ⊢ (𝑘 ∈ (ℤ≥‘𝐾) → (𝜑 → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
83	82	a2d 26	. . 3 ⊢ (𝑘 ∈ (ℤ≥‘𝐾) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
84	5, 7, 9, 11, 55, 83	uzind4 9656	. 2 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
85	3, 84	mpcom 36	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ‘cfv 5255  (class class class)co 5919  1c1 7875   + caddc 7877   − cmin 8192  ℤcz 9320  ℤ_≥cuz 9595  ...cfz 10077  seqcseq 10521

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-fz 10078  df-seqfrec 10522

This theorem is referenced by:  bcval5  10837  lgsne0  15195