Theorem seq3f1o 10976

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 10450	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
3	2	iftrued 3647	. . . . 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 10442	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
6		fveq2 5698	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
7	6	eleq1d 2307	. . . . . . 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 2623	. . . . . . . 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 5642	. . . . . . . . . 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 5846	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
15		elfzuz 10442	. . . . . . . 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 2934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑘)) ∈ 𝑆)
18	4, 17	eqeltrd 2315	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆)
19		breq1 4134	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
20		2fveq3 5703	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑘)))
21	19, 20	ifbieq1d 3663	. . . . . 6 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
22		eqid 2238	. . . . . 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 5784	. . . . 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 602	. . . 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 2282	. . 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 5698	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐺‘𝑏) = (𝐺‘(𝐹‘𝑥)))
30	29	eleq1d 2307	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐺‘𝑏) ∈ 𝑆 ↔ (𝐺‘(𝐹‘𝑥)) ∈ 𝑆))
31		fveq2 5698	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (𝐺‘𝑥) = (𝐺‘𝑏))
32	31	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑏) ∈ 𝑆))
33	32	cbvralv 2786	. . . . . . . . 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 492	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → ∀𝑏 ∈ (ℤ≥‘𝑀)(𝐺‘𝑏) ∈ 𝑆)
36	13	ad2antrr 492	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
37		eluzel2 9943	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
38	1, 37	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
39	38	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ∈ ℤ)
40		eluzelz 9948	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
41	1, 40	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42	41	ad2antrr 492	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑁 ∈ ℤ)
43		eluzelz 9948	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
44	43	ad2antlr 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ ℤ)
45		eluzle 9951	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑀 ≤ 𝑥)
46	45	ad2antlr 493	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑀 ≤ 𝑥)
47		simpr 110	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ≤ 𝑁)
48		elfz4 10439	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (𝑀 ≤ 𝑥 ∧ 𝑥 ≤ 𝑁)) → 𝑥 ∈ (𝑀...𝑁))
49	39, 42, 44, 46, 47, 48	syl32anc 1286	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (𝑀...𝑁))
50	36, 49	ffvelcdmd 5847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐹‘𝑥) ∈ (𝑀...𝑁))
51		elfzuz 10442	. . . . . . . 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 2934	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐺‘(𝐹‘𝑥)) ∈ 𝑆)
54		fveq2 5698	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
55	54	eleq1d 2307	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
56		uzid 9953	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
57	38, 56	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
58	55, 9, 57	rspcdva 2934	. . . . . . 7 ⊢ (𝜑 → (𝐺‘𝑀) ∈ 𝑆)
59	58	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → (𝐺‘𝑀) ∈ 𝑆)
60	41	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
61		zdcle 9733	. . . . . . 7 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
62	43, 60, 61	syl2an2 602	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → DECID 𝑥 ≤ 𝑁)
63	53, 59, 62	ifcldadc 3670	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆)
64		breq1 4134	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
65		2fveq3 5703	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
66	64, 65	ifbieq1d 3663	. . . . . 6 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
67	66, 22	fvmptg 5784	. . . . 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 415	. . . 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 2315	. . 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 10938	. 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 4229	. . 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 10975	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
76	71, 75	eqtrd 2271	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ifcif 3638   class class class wbr 4131   ↦ cmpt 4193  ⟶wf 5376  –1-1-onto→wf1o 5379  ‘cfv 5380  (class class class)co 6088   ≤ cle 8368  ℤcz 9656  ℤ_≥cuz 9938  ...cfz 10429  seqcseq 10906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-1o 6691  df-er 6811  df-en 7028  df-fin 7030  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-inn 9315  df-n0 9576  df-z 9657  df-uz 9939  df-fz 10430  df-fzo 10568  df-seqfrec 10907

This theorem is used by:  summodclem3  12173  prodmodclem3  12368