Theorem seqof 12858

Description: Distribute function operation through a sequence. Note that 𝐺(𝑧) is an implicit function on 𝑧. (Contributed by Mario Carneiro, 3-Mar-2015.)

Hypotheses

Ref	Expression
seqof.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
seqof.2	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
seqof.3	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))

Assertion

Ref	Expression
seqof	⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Distinct variable groups:   𝑥,𝑧,𝐴   𝑥,𝐹,𝑧   𝑥,𝐺   𝑥,𝑀,𝑧   𝑥,𝑁,𝑧   𝑥, + ,𝑧   𝜑,𝑥,𝑧

Allowed substitution hints:   𝐺(𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem seqof

Dummy variables 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		seqof.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
2		fvex 6201	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
3	2	rgenw 2924	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V
4		eqid 2622	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))
5	4	fnmpt 6020	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V → (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴)
6	3, 5	mp1i 13	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴)
7		seqof.3	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
8	7	fneq1d 5981	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) Fn 𝐴 ↔ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴))
9	6, 8	mpbird 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) Fn 𝐴)
10		fvex 6201	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		fneq1 5979	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹‘𝑥) Fn 𝐴))
12	10, 11	elab 3350	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝐹‘𝑥) Fn 𝐴)
13	9, 12	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
14		simprl 794	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15		simprr 796	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16		seqof.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝐴 ∈ 𝑉)
18		inidm 3822	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	14, 15, 17, 17, 18	offn 6908	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → (𝑥 ∘𝑓 + 𝑦) Fn 𝐴)
20	19	ex 450	. . . . . . 7 ⊢ (𝜑 → ((𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴) → (𝑥 ∘𝑓 + 𝑦) Fn 𝐴))
21		vex 3203	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22		fneq1 5979	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴))
23	21, 22	elab 3350	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24		vex 3203	. . . . . . . . 9 ⊢ 𝑦 ∈ V
25		fneq1 5979	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
26	24, 25	elab 3350	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
27	23, 26	anbi12i 733	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴))
28		ovex 6678	. . . . . . . 8 ⊢ (𝑥 ∘𝑓 + 𝑦) ∈ V
29		fneq1 5979	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∘𝑓 + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥 ∘𝑓 + 𝑦) Fn 𝐴))
30	28, 29	elab 3350	. . . . . . 7 ⊢ ((𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝑥 ∘𝑓 + 𝑦) Fn 𝐴)
31	20, 27, 30	3imtr4g 285	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴}))
32	31	imp 445	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
33	1, 13, 32	seqcl 12821	. . . 4 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
34		fvex 6201	. . . . 5 ⊢ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V
35		fneq1 5979	. . . . 5 ⊢ (𝑧 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴))
36	34, 35	elab 3350	. . . 4 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
37	33, 36	sylib 208	. . 3 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
38		dffn5 6241	. . 3 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
39	37, 38	sylib 208	. 2 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
40		fveq1 6190	. . . . . 6 ⊢ (𝑤 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑤‘𝑧) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
41		eqid 2622	. . . . . 6 ⊢ (𝑤 ∈ V ↦ (𝑤‘𝑧)) = (𝑤 ∈ V ↦ (𝑤‘𝑧))
42		fvex 6201	. . . . . 6 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) ∈ V
43	40, 41, 42	fvmpt 6282	. . . . 5 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
44	34, 43	mp1i 13	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
45	32	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
46	13	adantlr 751	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
47	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑁 ∈ (ℤ≥‘𝑀))
48		eqidd 2623	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘𝑧) = (𝑥‘𝑧))
49		eqidd 2623	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘𝑧) = (𝑦‘𝑧))
50	14, 15, 17, 17, 18, 48, 49	ofval 6906	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∘𝑓 + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
51	50	an32s 846	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑥 ∘𝑓 + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
52		fveq1 6190	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∘𝑓 + 𝑦) → (𝑤‘𝑧) = ((𝑥 ∘𝑓 + 𝑦)‘𝑧))
53		fvex 6201	. . . . . . . . 9 ⊢ ((𝑥 ∘𝑓 + 𝑦)‘𝑧) ∈ V
54	52, 41, 53	fvmpt 6282	. . . . . . . 8 ⊢ ((𝑥 ∘𝑓 + 𝑦) ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝑥 ∘𝑓 + 𝑦)) = ((𝑥 ∘𝑓 + 𝑦)‘𝑧))
55	28, 54	ax-mp 5	. . . . . . 7 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝑥 ∘𝑓 + 𝑦)) = ((𝑥 ∘𝑓 + 𝑦)‘𝑧)
56		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑧) = (𝑥‘𝑧))
57		fvex 6201	. . . . . . . . . 10 ⊢ (𝑥‘𝑧) ∈ V
58	56, 41, 57	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧))
59	21, 58	ax-mp 5	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧)
60		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤‘𝑧) = (𝑦‘𝑧))
61		fvex 6201	. . . . . . . . . 10 ⊢ (𝑦‘𝑧) ∈ V
62	60, 41, 61	fvmpt 6282	. . . . . . . . 9 ⊢ (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧))
63	24, 62	ax-mp 5	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧)
64	59, 63	oveq12i 6662	. . . . . . 7 ⊢ (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)) = ((𝑥‘𝑧) + (𝑦‘𝑧))
65	51, 55, 64	3eqtr4g 2681	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝑥 ∘𝑓 + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)))
66	27, 65	sylan2b 492	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝑥 ∘𝑓 + 𝑦)) = (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)))
67		fveq1 6190	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝑤‘𝑧) = ((𝐹‘𝑥)‘𝑧))
68		fvex 6201	. . . . . . . 8 ⊢ ((𝐹‘𝑥)‘𝑧) ∈ V
69	67, 41, 68	fvmpt 6282	. . . . . . 7 ⊢ ((𝐹‘𝑥) ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = ((𝐹‘𝑥)‘𝑧))
70	10, 69	ax-mp 5	. . . . . 6 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = ((𝐹‘𝑥)‘𝑧)
71	7	adantlr 751	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
72	71	fveq1d 6193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧))
73		simplr 792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧 ∈ 𝐴)
74	4	fvmpt2 6291	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑥) ∈ V) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
75	73, 2, 74	sylancl 694	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
76	72, 75	eqtrd 2656	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = (𝐺‘𝑥))
77	70, 76	syl5eq 2668	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = (𝐺‘𝑥))
78	45, 46, 47, 66, 77	seqhomo 12848	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
79	44, 78	eqtr3d 2658	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
80	79	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
81	39, 80	eqtrd 2656	1 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895  ℤ_≥cuz 11687  ...cfz 12326  seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  seqof2  12859  mtest  24158  pserulm  24176  knoppcnlem7  32489