Theorem seq3f1o 10308

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 9839	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
3	2	iftrued 3486	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) = (𝐺‘(𝐹‘𝑘)))
4	3	adantl 275	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) = (𝐺‘(𝐹‘𝑘)))
5		elfzuz 9833	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
6		fveq2 5429	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
7	6	eleq1d 2209	. . . . . . 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 2508	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
10	9	adantr 274	. . . . . . 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 5375	. . . . . . . . . 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	ffvelrnda 5563	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
15		elfzuz 9833	. . . . . . . 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 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑘)) ∈ 𝑆)
18	4, 17	eqeltrd 2217	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆)
19		breq1 3940	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
20		2fveq3 5434	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑘)))
21	19, 20	ifbieq1d 3499	. . . . . 6 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
22		eqid 2140	. . . . . 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 5505	. . . . 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 584	. . . 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 2184	. . 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 109	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ (ℤ≥‘𝑀))
29		fveq2 5429	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐺‘𝑏) = (𝐺‘(𝐹‘𝑥)))
30	29	eleq1d 2209	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐺‘𝑏) ∈ 𝑆 ↔ (𝐺‘(𝐹‘𝑥)) ∈ 𝑆))
31		fveq2 5429	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (𝐺‘𝑥) = (𝐺‘𝑏))
32	31	eleq1d 2209	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑏) ∈ 𝑆))
33	32	cbvralv 2657	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆 ↔ ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
34	9, 33	sylib 121	. . . . . . . 8 ⊢ (𝜑 → ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
35	34	ad2antrr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
36	13	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
37		eluzel2 9355	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
38	1, 37	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
39	38	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ∈ ℤ)
40		eluzelz 9359	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
41	1, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42	41	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑁 ∈ ℤ)
43		eluzelz 9359	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
44	43	ad2antlr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ ℤ)
45		eluzle 9362	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑥)
46	45	ad2antlr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ≤ 𝑥)
47		simpr 109	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ≤ 𝑁)
48		elfz4 9830	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (𝑀 ≤ 𝑥 ∧ 𝑥 ≤ 𝑁)) → 𝑥 ∈ (𝑀...𝑁))
49	39, 42, 44, 46, 47, 48	syl32anc 1225	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (𝑀...𝑁))
50	36, 49	ffvelrnd 5564	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐹‘𝑥) ∈ (𝑀...𝑁))
51		elfzuz 9833	. . . . . . . 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 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐺‘(𝐹‘𝑥)) ∈ 𝑆)
54		fveq2 5429	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
55	54	eleq1d 2209	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
56		uzid 9364	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
57	38, 56	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
58	55, 9, 57	rspcdva 2798	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝑆)
59	58	ad2antrr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → (𝐺‘𝑀) ∈ 𝑆)
60	41	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
61		zdcle 9151	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
62	43, 60, 61	syl2an2 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → DECID 𝑥 ≤ 𝑁)
63	53, 59, 62	ifcldadc 3506	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆)
64		breq1 3940	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
65		2fveq3 5434	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
66	64, 65	ifbieq1d 3499	. . . . . 6 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
67	66, 22	fvmptg 5505	. . . . 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 409	. . . 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 2217	. . 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 10275	. 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 4032	. . 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 10307	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
76	71, 75	eqtrd 2173	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  DECID wdc 820   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ifcif 3479   class class class wbr 3937   ↦ cmpt 3997  ⟶wf 5127  –1-1-onto→wf1o 5130  ‘cfv 5131  (class class class)co 5782   ≤ cle 7825  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-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-dc 821  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-if 3480  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-1o 6321  df-er 6437  df-en 6643  df-fin 6645  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-fzo 9951  df-seqfrec 10250

This theorem is referenced by:  summodclem3  11181  prodmodclem3  11376