Theorem seq3f1o 10626

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 10120	. . . . . 6 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
3	2	iftrued 3569	. . . . 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 10113	. . . . 5 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
6		fveq2 5561	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑘) → (𝐺‘𝑥) = (𝐺‘(𝐹‘𝑘)))
7	6	eleq1d 2265	. . . . . . 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 2570	. . . . . . . 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 5507	. . . . . . . . . 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 5700	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...𝑁))
15		elfzuz 10113	. . . . . . . 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 2873	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝐹‘𝑘)) ∈ 𝑆)
18	4, 17	eqeltrd 2273	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆)
19		breq1 4037	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
20		2fveq3 5566	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑘)))
21	19, 20	ifbieq1d 3584	. . . . . 6 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝐹‘𝑘)), (𝐺‘𝑀)))
22		eqid 2196	. . . . . 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 5640	. . . . 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 2240	. . 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 5561	. . . . . . . 8 ⊢ (𝑏 = (𝐹‘𝑥) → (𝐺‘𝑏) = (𝐺‘(𝐹‘𝑥)))
30	29	eleq1d 2265	. . . . . . 7 ⊢ (𝑏 = (𝐹‘𝑥) → ((𝐺‘𝑏) ∈ 𝑆 ↔ (𝐺‘(𝐹‘𝑥)) ∈ 𝑆))
31		fveq2 5561	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑏 → (𝐺‘𝑥) = (𝐺‘𝑏))
32	31	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑏) ∈ 𝑆))
33	32	cbvralv 2729	. . . . . . . . 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 9623	. . . . . . . . . . . 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 9627	. . . . . . . . . . . 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 9627	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
44	43	ad2antlr 489	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ ℤ)
45		eluzle 9630	. . . . . . . . . . 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 10110	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑥 ∈ ℤ) ∧ (𝑀 ≤ 𝑥 ∧ 𝑥 ≤ 𝑁)) → 𝑥 ∈ (𝑀...𝑁))
49	39, 42, 44, 46, 47, 48	syl32anc 1257	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (𝑀...𝑁))
50	36, 49	ffvelcdmd 5701	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐹‘𝑥) ∈ (𝑀...𝑁))
51		elfzuz 10113	. . . . . . . 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 2873	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐺‘(𝐹‘𝑥)) ∈ 𝑆)
54		fveq2 5561	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → (𝐺‘𝑥) = (𝐺‘𝑀))
55	54	eleq1d 2265	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
56		uzid 9632	. . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
57	38, 56	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀))
58	55, 9, 57	rspcdva 2873	. . . . . . 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 9419	. . . . . . 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 3591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆)
64		breq1 4037	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
65		2fveq3 5566	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝐹‘𝑎)) = (𝐺‘(𝐹‘𝑥)))
66	64, 65	ifbieq1d 3584	. . . . . 6 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
67	66, 22	fvmptg 5640	. . . . 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 2273	. . 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 10588	. 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 4130	. . 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 10625	. 2 ⊢ (𝜑 → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝐹‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
76	71, 75	eqtrd 2229	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ifcif 3562   class class class wbr 4034   ↦ cmpt 4095  ⟶wf 5255  –1-1-onto→wf1o 5258  ‘cfv 5259  (class class class)co 5925   ≤ cle 8079  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  seqcseq 10556

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-dc 836  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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-fz 10101  df-fzo 10235  df-seqfrec 10557

This theorem is referenced by:  summodclem3  11562  prodmodclem3  11757