Theorem seq3z 10315

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 9834	. . 3 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝐾))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
4		fveqeq2 5438	. . . 4 ⊢ (𝑤 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
5	4	imbi2d 229	. . 3 ⊢ (𝑤 = 𝐾 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)))
6		fveqeq2 5438	. . . 4 ⊢ (𝑤 = 𝑘 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍))
7	6	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)))
8		fveqeq2 5438	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
9	8	imbi2d 229	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)))
10		fveqeq2 5438	. . . 4 ⊢ (𝑤 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝑤) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝑁) = 𝑍))
11	10	imbi2d 229	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → (seq𝑀( + , 𝐹)‘𝑤) = 𝑍) ↔ (𝜑 → (seq𝑀( + , 𝐹)‘𝑁) = 𝑍)))
12		elfzuz 9833	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ≥‘𝑀))
13	1, 12	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
14		eluzelz 9359	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝐾 ∈ ℤ)
15	13, 14	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ ℤ)
16		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝑥 ∈ (ℤ≥‘𝐾))
17	13	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀))
18		uztrn 9366	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ (ℤ≥‘𝑀))
19	16, 17, 18	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝐾)) → 𝑥 ∈ (ℤ≥‘𝑀))
20		seq3homo.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
21	19, 20	syldan 280	. . . . . . . 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 10264	. . . . . . 7 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = (𝐹‘𝐾))
24		seqz.7	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐾) = 𝑍)
25	23, 24	eqtrd 2173	. . . . . 6 ⊢ (𝜑 → (seq𝐾( + , 𝐹)‘𝐾) = 𝑍)
26		seqeq1 10252	. . . . . . . 8 ⊢ (𝐾 = 𝑀 → seq𝐾( + , 𝐹) = seq𝑀( + , 𝐹))
27	26	fveq1d 5431	. . . . . . 7 ⊢ (𝐾 = 𝑀 → (seq𝐾( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
28	27	eqeq1d 2149	. . . . . 6 ⊢ (𝐾 = 𝑀 → ((seq𝐾( + , 𝐹)‘𝐾) = 𝑍 ↔ (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
29	25, 28	syl5ibcom 154	. . . . 5 ⊢ (𝜑 → (𝐾 = 𝑀 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
30		eluzel2 9355	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
31	13, 30	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
32	31	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
33		simpr 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → 𝐾 ∈ (ℤ≥‘(𝑀 + 1)))
34	20	adantlr 469	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
35	22	adantlr 469	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
36	32, 33, 34, 35	seq3m1 10272	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)))
37	24	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐹‘𝐾) = 𝑍)
38	37	oveq2d 5798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + (𝐹‘𝐾)) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
39		oveq1 5789	. . . . . . . . 9 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘(𝐾 − 1)) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍))
40	39	eqeq1d 2149	. . . . . . . 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 2508	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍)
43	42	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑍)
44		eqid 2140	. . . . . . . . . 10 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
45	44, 32, 34, 35	seqf 10265	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → seq𝑀( + , 𝐹):(ℤ≥‘𝑀)⟶𝑆)
46		eluzp1m1 9373	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ≥‘𝑀))
47	31, 46	sylan 281	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐾 − 1) ∈ (ℤ≥‘𝑀))
48	45, 47	ffvelrnd 5564	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘(𝐾 − 1)) ∈ 𝑆)
49	40, 43, 48	rspcdva 2798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → ((seq𝑀( + , 𝐹)‘(𝐾 − 1)) + 𝑍) = 𝑍)
50	36, 38, 49	3eqtrd 2177	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ (ℤ≥‘(𝑀 + 1))) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
51	50	ex 114	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ (ℤ≥‘(𝑀 + 1)) → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
52		uzp1 9383	. . . . . 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 708	. . . 4 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍)
55	54	a1i 9	. . 3 ⊢ (𝐾 ∈ ℤ → (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = 𝑍))
56		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝑘 ∈ (ℤ≥‘𝐾))
57	13	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝐾 ∈ (ℤ≥‘𝑀))
58		uztrn 9366	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
59	56, 57, 58	syl2anc 409	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → 𝑘 ∈ (ℤ≥‘𝑀))
60	20	adantlr 469	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
61	22	adantlr 469	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
62	59, 60, 61	seq3p1 10266	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
63	62	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))))
64		simpr 109	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘𝑘) = 𝑍)
65	64	oveq1d 5797	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → ((seq𝑀( + , 𝐹)‘𝑘) + (𝐹‘(𝑘 + 1))) = (𝑍 + (𝐹‘(𝑘 + 1))))
66		oveq2 5790	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑍 + 𝑥) = (𝑍 + (𝐹‘(𝑘 + 1))))
67	66	eqeq1d 2149	. . . . . . . . 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 2508	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍)
70	69	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ∀𝑥 ∈ 𝑆 (𝑍 + 𝑥) = 𝑍)
71		fveq2 5429	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑘 + 1) → (𝐹‘𝑥) = (𝐹‘(𝑘 + 1)))
72	71	eleq1d 2209	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 + 1) → ((𝐹‘𝑥) ∈ 𝑆 ↔ (𝐹‘(𝑘 + 1)) ∈ 𝑆))
73	20	ralrimiva 2508	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
74	73	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐹‘𝑥) ∈ 𝑆)
75		peano2uz 9405	. . . . . . . . . . 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 2798	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝐹‘(𝑘 + 1)) ∈ 𝑆)
78	67, 70, 77	rspcdva 2798	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
79	78	adantr 274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (𝑍 + (𝐹‘(𝑘 + 1))) = 𝑍)
80	63, 65, 79	3eqtrd 2177	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) ∧ (seq𝑀( + , 𝐹)‘𝑘) = 𝑍) → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍)
81	80	ex 114	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝐾)) → ((seq𝑀( + , 𝐹)‘𝑘) = 𝑍 → (seq𝑀( + , 𝐹)‘(𝑘 + 1)) = 𝑍))
82	81	expcom 115	. . . 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 9410	. 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 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ‘cfv 5131  (class class class)co 5782  1c1 7645   + caddc 7647   − cmin 7957  ℤcz 9078  ℤ_≥cuz 9350  ...cfz 9821  seqcseq 10249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-seqfrec 10250

This theorem is referenced by:  bcval5  10541