Theorem seqof 12720

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 6113	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
3	2	rgenw 2908	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V
4		eqid 2610	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))
5	4	fnmpt 5933	. . . . . . . 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 5895	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) Fn 𝐴 ↔ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴))
9	6, 8	mpbird 246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) Fn 𝐴)
10		fvex 6113	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		fneq1 5893	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹‘𝑥) Fn 𝐴))
12	10, 11	elab 3319	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝐹‘𝑥) Fn 𝐴)
13	9, 12	sylibr 223	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
14		simprl 790	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15		simprr 792	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16		seqof.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝐴 ∈ 𝑉)
18		inidm 3784	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	14, 15, 17, 17, 18	offn 6806	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → (𝑥 ∘𝑓 + 𝑦) Fn 𝐴)
20	19	ex 449	. . . . . . 7 ⊢ (𝜑 → ((𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴) → (𝑥 ∘𝑓 + 𝑦) Fn 𝐴))
21		vex 3176	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22		fneq1 5893	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴))
23	21, 22	elab 3319	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24		vex 3176	. . . . . . . . 9 ⊢ 𝑦 ∈ V
25		fneq1 5893	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
26	24, 25	elab 3319	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
27	23, 26	anbi12i 729	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴))
28		ovex 6577	. . . . . . . 8 ⊢ (𝑥 ∘𝑓 + 𝑦) ∈ V
29		fneq1 5893	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∘𝑓 + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥 ∘𝑓 + 𝑦) Fn 𝐴))
30	28, 29	elab 3319	. . . . . . 7 ⊢ ((𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝑥 ∘𝑓 + 𝑦) Fn 𝐴)
31	20, 27, 30	3imtr4g 284	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴}))
32	31	imp 444	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
33	1, 13, 32	seqcl 12683	. . . 4 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
34		fvex 6113	. . . . 5 ⊢ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ V
35		fneq1 5893	. . . . 5 ⊢ (𝑧 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴))
36	34, 35	elab 3319	. . . 4 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
37	33, 36	sylib 207	. . 3 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴)
38		dffn5 6151	. . 3 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
39	37, 38	sylib 207	. 2 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)))
40		fveq1 6102	. . . . . 6 ⊢ (𝑤 = (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) → (𝑤‘𝑧) = ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧))
41		eqid 2610	. . . . . 6 ⊢ (𝑤 ∈ V ↦ (𝑤‘𝑧)) = (𝑤 ∈ V ↦ (𝑤‘𝑧))
42		fvex 6113	. . . . . 6 ⊢ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) ∈ V
43	40, 41, 42	fvmpt 6191	. . . . 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 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘𝑓 + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
46	13	adantlr 747	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
47	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑁 ∈ (ℤ≥‘𝑀))
48		eqidd 2611	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘𝑧) = (𝑥‘𝑧))
49		eqidd 2611	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘𝑧) = (𝑦‘𝑧))
50	14, 15, 17, 17, 18, 48, 49	ofval 6804	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∘𝑓 + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
51	50	an32s 842	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑥 ∘𝑓 + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
52		fveq1 6102	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∘𝑓 + 𝑦) → (𝑤‘𝑧) = ((𝑥 ∘𝑓 + 𝑦)‘𝑧))
53		fvex 6113	. . . . . . . . 9 ⊢ ((𝑥 ∘𝑓 + 𝑦)‘𝑧) ∈ V
54	52, 41, 53	fvmpt 6191	. . . . . . . 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 6102	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑧) = (𝑥‘𝑧))
57		fvex 6113	. . . . . . . . . 10 ⊢ (𝑥‘𝑧) ∈ V
58	56, 41, 57	fvmpt 6191	. . . . . . . . 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 6102	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤‘𝑧) = (𝑦‘𝑧))
61		fvex 6113	. . . . . . . . . 10 ⊢ (𝑦‘𝑧) ∈ V
62	60, 41, 61	fvmpt 6191	. . . . . . . . 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 6561	. . . . . . 7 ⊢ (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)) = ((𝑥‘𝑧) + (𝑦‘𝑧))
65	51, 55, 64	3eqtr4g 2669	. . . . . 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 491	. . . . 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 6102	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝑤‘𝑧) = ((𝐹‘𝑥)‘𝑧))
68		fvex 6113	. . . . . . . 8 ⊢ ((𝐹‘𝑥)‘𝑧) ∈ V
69	67, 41, 68	fvmpt 6191	. . . . . . 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 747	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
72	71	fveq1d 6105	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧))
73		simplr 788	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧 ∈ 𝐴)
74	4	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑥) ∈ V) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
75	73, 2, 74	sylancl 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
76	72, 75	eqtrd 2644	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = (𝐺‘𝑥))
77	70, 76	syl5eq 2656	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = (𝐺‘𝑥))
78	45, 46, 47, 66, 77	seqhomo 12710	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
79	44, 78	eqtr3d 2646	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
80	79	mpteq2dva 4672	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘𝑓 + , 𝐹)‘𝑁)‘𝑧)) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
81	39, 80	eqtrd 2644	1 ⊢ (𝜑 → (seq𝑀( ∘𝑓 + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ↦ cmpt 4643   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ∘_𝑓 cof 6793  ℤ_≥cuz 11563  ...cfz 12197  seqcseq 12663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664

This theorem is referenced by:  seqof2  12721  mtest  23962  pserulm  23980  knoppcnlem7  31659