Theorem seq3id2 10618

Description: The last few partial sums of a sequence that ends with all zeroes (or any element which is a right-identity for +) are all the same. (Contributed by Mario Carneiro, 13-Jul-2013.) (Revised by Jim Kingdon, 12-Nov-2022.)

Hypotheses

Ref	Expression
seqid2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)
seqid2.2	⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
seqid2.3	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
seqid2.4	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
seqid2.5	⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)
seq3id2.f	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
seq3id2.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)

Assertion

Ref	Expression
seq3id2	⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))

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

Allowed substitution hint:   𝑍(𝑦)

Proof of Theorem seq3id2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqid2.3	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝐾))
2		eluzfz2 10107	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → 𝑁 ∈ (𝐾...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝐾...𝑁))
4		eleq1 2259	. . . . . 6 ⊢ (𝑥 = 𝐾 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝐾 ∈ (𝐾...𝑁)))
5		fveq2 5558	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝐾))
6	5	eqeq2d 2208	. . . . . 6 ⊢ (𝑥 = 𝐾 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))
7	4, 6	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝐾 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
8	7	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝐾 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾)))))
9		eleq1 2259	. . . . . 6 ⊢ (𝑥 = 𝑛 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑛 ∈ (𝐾...𝑁)))
10		fveq2 5558	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑛))
11	10	eqeq2d 2208	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))
12	9, 11	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝑛 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
13	12	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝑛 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)))))
14		eleq1 2259	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → (𝑥 ∈ (𝐾...𝑁) ↔ (𝑛 + 1) ∈ (𝐾...𝑁)))
15		fveq2 5558	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))
16	15	eqeq2d 2208	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
17	14, 16	imbi12d 234	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)))))
18	17	imbi2d 230	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
19		eleq1 2259	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑥 ∈ (𝐾...𝑁) ↔ 𝑁 ∈ (𝐾...𝑁)))
20		fveq2 5558	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑥) = (seq𝑀( + , 𝐹)‘𝑁))
21	20	eqeq2d 2208	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥) ↔ (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
22	19, 21	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥)) ↔ (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
23	22	imbi2d 230	. . . 4 ⊢ (𝑥 = 𝑁 → ((𝜑 → (𝑥 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑥))) ↔ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))))
24		eqidd 2197	. . . . 5 ⊢ (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))
25	24	2a1i 27	. . . 4 ⊢ (𝐾 ∈ ℤ → (𝜑 → (𝐾 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝐾))))
26		peano2fzr 10112	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁)) → 𝑛 ∈ (𝐾...𝑁))
27	26	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (𝐾...𝑁))
28	27	expr 375	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑛 ∈ (𝐾...𝑁)))
29	28	imim1d 75	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ≥‘𝐾)) → ((𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛)) → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))))
30		oveq1 5929	. . . . . 6 ⊢ ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
31		fveqeq2 5567	. . . . . . . . . 10 ⊢ (𝑥 = (𝑛 + 1) → ((𝐹‘𝑥) = 𝑍 ↔ (𝐹‘(𝑛 + 1)) = 𝑍))
32		seqid2.5	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝐾 + 1)...𝑁)) → (𝐹‘𝑥) = 𝑍)
33	32	ralrimiva 2570	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑥) = 𝑍)
34	33	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ∀𝑥 ∈ ((𝐾 + 1)...𝑁)(𝐹‘𝑥) = 𝑍)
35		eluzp1p1 9627	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
36	35	ad2antrl 490	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)))
37		elfzuz3 10097	. . . . . . . . . . . 12 ⊢ ((𝑛 + 1) ∈ (𝐾...𝑁) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
38	37	ad2antll 491	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑁 ∈ (ℤ≥‘(𝑛 + 1)))
39		elfzuzb 10094	. . . . . . . . . . 11 ⊢ ((𝑛 + 1) ∈ ((𝐾 + 1)...𝑁) ↔ ((𝑛 + 1) ∈ (ℤ≥‘(𝐾 + 1)) ∧ 𝑁 ∈ (ℤ≥‘(𝑛 + 1))))
40	36, 38, 39	sylanbrc 417	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝑛 + 1) ∈ ((𝐾 + 1)...𝑁))
41	31, 34, 40	rspcdva 2873	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (𝐹‘(𝑛 + 1)) = 𝑍)
42	41	oveq2d 5938	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
43		oveq1 5929	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → (𝑥 + 𝑍) = ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍))
44		id 19	. . . . . . . . . . 11 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → 𝑥 = (seq𝑀( + , 𝐹)‘𝐾))
45	43, 44	eqeq12d 2211	. . . . . . . . . 10 ⊢ (𝑥 = (seq𝑀( + , 𝐹)‘𝐾) → ((𝑥 + 𝑍) = 𝑥 ↔ ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾)))
46		seqid2.1	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 + 𝑍) = 𝑥)
47	46	ralrimiva 2570	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (𝑥 + 𝑍) = 𝑥)
48		seqid2.4	. . . . . . . . . 10 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) ∈ 𝑆)
49	45, 47, 48	rspcdva 2873	. . . . . . . . 9 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
50	49	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) + 𝑍) = (seq𝑀( + , 𝐹)‘𝐾))
51	42, 50	eqtr2d 2230	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘𝐾) = ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))))
52		simprl 529	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝐾))
53		seqid2.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ∈ (ℤ≥‘𝑀))
54	53	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝐾 ∈ (ℤ≥‘𝑀))
55		uztrn 9618	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ≥‘𝐾) ∧ 𝐾 ∈ (ℤ≥‘𝑀)) → 𝑛 ∈ (ℤ≥‘𝑀))
56	52, 54, 55	syl2anc 411	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → 𝑛 ∈ (ℤ≥‘𝑀))
57		seq3id2.f	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
58	57	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑥) ∈ 𝑆)
59		seq3id2.cl	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
60	59	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
61	56, 58, 60	seq3p1 10557	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → (seq𝑀( + , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1))))
62	51, 61	eqeq12d 2211	. . . . . 6 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1)) ↔ ((seq𝑀( + , 𝐹)‘𝐾) + (𝐹‘(𝑛 + 1))) = ((seq𝑀( + , 𝐹)‘𝑛) + (𝐹‘(𝑛 + 1)))))
63	30, 62	imbitrrid 156	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (ℤ≥‘𝐾) ∧ (𝑛 + 1) ∈ (𝐾...𝑁))) → ((seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))
64	29, 63	animpimp2impd 559	. . . 4 ⊢ (𝑛 ∈ (ℤ≥‘𝐾) → ((𝜑 → (𝑛 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑛))) → (𝜑 → ((𝑛 + 1) ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘(𝑛 + 1))))))
65	8, 13, 18, 23, 25, 64	uzind4 9662	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝐾) → (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))))
66	1, 65	mpcom 36	. 2 ⊢ (𝜑 → (𝑁 ∈ (𝐾...𝑁) → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁)))
67	3, 66	mpd 13	1 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝐾) = (seq𝑀( + , 𝐹)‘𝑁))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ‘cfv 5258  (class class class)co 5922  1c1 7880   + caddc 7882  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  seqcseq 10539

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540

This theorem is referenced by:  seq3coll  10934  fsum3cvg  11543  fproddccvg  11737  lgsdilem2  15277