Theorem seq3f1o 10506

Description: Rearrange a sum via an arbitrary bijection on (𝑀...𝑁). (Contributed by Mario Carneiro, 27-Feb-2014.) (Revised by Jim Kingdon, 3-Nov-2022.)

Hypotheses

Ref	Expression
iseqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqf1o.6	⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
iseqf1o.h	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆)
iseqf1o.8	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))

Assertion

Ref	Expression
seq3f1o	⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑘,𝑀   𝑥,𝑁,𝑦,𝑧   𝑘,𝑁   𝑥,𝐺,𝑦,𝑧   𝑘,𝐺   𝑥,𝐹,𝑦,𝑧   𝑘,𝐹   𝑥,𝐻,𝑦,𝑘   𝑥,𝑆,𝑦,𝑧   𝑆,𝑘   𝑥, + ,𝑦,𝑧   + ,𝑘   𝜑,𝑥,𝑦,𝑧   𝜑,𝑘,𝑥,𝑦

Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem seq3f1o

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1o.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		elfzle2 10030	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
3	2	iftrued 3543	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) = (𝐺‘(𝐹‘𝑘)))
4	3	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) = (𝐺‘(𝐹‘𝑘)))
5		elfzuz 10023	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
6		fveq2 5517	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
7	6	eleq1d 2246	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑘) → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘(𝐹‘𝑘)) ∈ 𝑆))
8		iseqf1o.7	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
9	8	ralrimiva 2550	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
10	9	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
11		iseqf1o.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
12		f1of 5463	. . . . . . . . . 10 ⊢ (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
13	11, 12	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
14	13	ffvelcdmda 5653	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
15		elfzuz 10023	. . . . . . . 8 ⊢ ((𝐹‘𝑘) ∈ (𝑀...𝑁) → (𝐹‘𝑘) ∈ (ℤ≥‘𝑀))
16	14, 15	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (ℤ≥‘𝑀))
17	7, 10, 16	rspcdva 2848	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑘)) ∈ 𝑆)
18	4, 17	eqeltrd 2254	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆)
19		breq1 4008	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
20		2fveq3 5522	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑘)))
21	19, 20	ifbieq1d 3558	. . . . . 6 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
22		eqid 2177	. . . . . 6 ⊢ (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))) = (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))
23	21, 22	fvmptg 5594	. . . . 5 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑘) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
24	5, 18, 23	syl2an2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑘) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
25		iseqf1o.8	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = (𝐺‘(𝐹‘𝑘)))
26	4, 24, 25	3eqtr4rd 2221	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐻‘𝑘) = ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑘))
27		iseqf1o.h	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐻‘𝑥) ∈ 𝑆)
28		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ (ℤ≥‘𝑀))
29		fveq2 5517	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐺‘𝑏) = (𝐺‘(𝐹‘𝑥)))
30	29	eleq1d 2246	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐺‘𝑏) ∈ 𝑆 ↔ (𝐺‘(𝐹‘𝑥)) ∈ 𝑆))
31		fveq2 5517	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (𝐺‘𝑥) = (𝐺‘𝑏))
32	31	eleq1d 2246	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑏) ∈ 𝑆))
33	32	cbvralv 2705	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆 ↔ ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
34	9, 33	sylib 122	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
35	34	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
36	13	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
37		eluzel2 9535	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
38	1, 37	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
39	38	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ∈ ℤ)
40		eluzelz 9539	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
41	1, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42	41	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑁 ∈ ℤ)
43		eluzelz 9539	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
44	43	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ ℤ)
45		eluzle 9542	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑥)
46	45	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ≤ 𝑥)
47		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ≤ 𝑁)
48		elfz4 10020	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (𝑀 ≤ 𝑥 ∧ 𝑥 ≤ 𝑁)) → 𝑥 ∈ (𝑀...𝑁))
49	39, 42, 44, 46, 47, 48	syl32anc 1246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (𝑀...𝑁))
50	36, 49	ffvelcdmd 5654	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐹‘𝑥) ∈ (𝑀...𝑁))
51		elfzuz 10023	. . . . . . . 8 ⊢ ((𝐹‘𝑥) ∈ (𝑀...𝑁) → (𝐹‘𝑥) ∈ (ℤ≥‘𝑀))
52	50, 51	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐹‘𝑥) ∈ (ℤ≥‘𝑀))
53	30, 35, 52	rspcdva 2848	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐺‘(𝐹‘𝑥)) ∈ 𝑆)
54		fveq2 5517	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
55	54	eleq1d 2246	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
56		uzid 9544	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
57	38, 56	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
58	55, 9, 57	rspcdva 2848	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝑆)
59	58	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → (𝐺‘𝑀) ∈ 𝑆)
60	41	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
61		zdcle 9331	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
62	43, 60, 61	syl2an2 594	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → DECID 𝑥 ≤ 𝑁)
63	53, 59, 62	ifcldadc 3565	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆)
64		breq1 4008	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
65		2fveq3 5522	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
66	64, 65	ifbieq1d 3558	. . . . . 6 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
67	66, 22	fvmptg 5594	. . . . 5 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑥) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
68	28, 63, 67	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑥) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
69	68, 63	eqeltrd 2254	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)))‘𝑥) ∈ 𝑆)
70		iseqf1o.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
71	1, 26, 27, 69, 70	seq3fveq 10473	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))))‘𝑁))
72		iseqf1o.2	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
73		iseqf1o.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
74	66	cbvmptv 4101	. . 3 ⊢ (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
75	70, 72, 73, 1, 11, 8, 74	seq3f1oleml 10505	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
76	71, 75	eqtrd 2210	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 834   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ifcif 3536   class class class wbr 4005   ↦ cmpt 4066  ⟶wf 5214  –1-1-onto→wf1o 5217  ‘cfv 5218  (class class class)co 5877   ≤ cle 7995  ℤcz 9255  ℤ_≥cuz 9530  ...cfz 10010  seqcseq 10447

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-1o 6419  df-er 6537  df-en 6743  df-fin 6745  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531  df-fz 10011  df-fzo 10145  df-seqfrec 10448

This theorem is referenced by:  summodclem3  11390  prodmodclem3  11585