Theorem seqof 13708

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 6769	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
3	2	rgenw 3075	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V
4		eqid 2738	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))
5	4	fnmpt 6557	. . . . . . . 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 6510	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) Fn 𝐴 ↔ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴))
9	6, 8	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) Fn 𝐴)
10		fvex 6769	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		fneq1 6508	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹‘𝑥) Fn 𝐴))
12	10, 11	elab 3602	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝐹‘𝑥) Fn 𝐴)
13	9, 12	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
14		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15		simprr 769	. . . . . . . . 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 4149	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	14, 15, 17, 17, 18	offn 7524	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → (𝑥 ∘f + 𝑦) Fn 𝐴)
20	19	ex 412	. . . . . . 7 ⊢ (𝜑 → ((𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴) → (𝑥 ∘f + 𝑦) Fn 𝐴))
21		vex 3426	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22		fneq1 6508	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴))
23	21, 22	elab 3602	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24		vex 3426	. . . . . . . . 9 ⊢ 𝑦 ∈ V
25		fneq1 6508	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
26	24, 25	elab 3602	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
27	23, 26	anbi12i 626	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴))
28		ovex 7288	. . . . . . . 8 ⊢ (𝑥 ∘f + 𝑦) ∈ V
29		fneq1 6508	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∘f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥 ∘f + 𝑦) Fn 𝐴))
30	28, 29	elab 3602	. . . . . . 7 ⊢ ((𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝑥 ∘f + 𝑦) Fn 𝐴)
31	20, 27, 30	3imtr4g 295	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴}))
32	31	imp 406	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
33	1, 13, 32	seqcl 13671	. . . 4 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
34		fvex 6769	. . . . 5 ⊢ (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35		fneq1 6508	. . . . 5 ⊢ (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
36	34, 35	elab 3602	. . . 4 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
37	33, 36	sylib 217	. . 3 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
38		dffn5 6810	. . 3 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
39	37, 38	sylib 217	. 2 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
40		fveq1 6755	. . . . . 6 ⊢ (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤‘𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41		eqid 2738	. . . . . 6 ⊢ (𝑤 ∈ V ↦ (𝑤‘𝑧)) = (𝑤 ∈ V ↦ (𝑤‘𝑧))
42		fvex 6769	. . . . . 6 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
43	40, 41, 42	fvmpt 6857	. . . . 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 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
46	13	adantlr 711	. . . . 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 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘𝑧) = (𝑥‘𝑧))
49		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘𝑧) = (𝑦‘𝑧))
50	14, 15, 17, 17, 18, 48, 49	ofval 7522	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
51	50	an32s 648	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
52		fveq1 6755	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∘f + 𝑦) → (𝑤‘𝑧) = ((𝑥 ∘f + 𝑦)‘𝑧))
53		fvex 6769	. . . . . . . . 9 ⊢ ((𝑥 ∘f + 𝑦)‘𝑧) ∈ V
54	52, 41, 53	fvmpt 6857	. . . . . . . 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 6755	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑧) = (𝑥‘𝑧))
57		fvex 6769	. . . . . . . . . 10 ⊢ (𝑥‘𝑧) ∈ V
58	56, 41, 57	fvmpt 6857	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧))
59	58	elv 3428	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧)
60		fveq1 6755	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤‘𝑧) = (𝑦‘𝑧))
61		fvex 6769	. . . . . . . . . 10 ⊢ (𝑦‘𝑧) ∈ V
62	60, 41, 61	fvmpt 6857	. . . . . . . . 9 ⊢ (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧))
63	62	elv 3428	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧)
64	59, 63	oveq12i 7267	. . . . . . 7 ⊢ (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)) = ((𝑥‘𝑧) + (𝑦‘𝑧))
65	51, 55, 64	3eqtr4g 2804	. . . . . 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 593	. . . . 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 6755	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝑤‘𝑧) = ((𝐹‘𝑥)‘𝑧))
68		fvex 6769	. . . . . . . 8 ⊢ ((𝐹‘𝑥)‘𝑧) ∈ V
69	67, 41, 68	fvmpt 6857	. . . . . . 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 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
72	71	fveq1d 6758	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧))
73		simplr 765	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧 ∈ 𝐴)
74	4	fvmpt2 6868	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑥) ∈ V) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
75	73, 2, 74	sylancl 585	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
76	72, 75	eqtrd 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = (𝐺‘𝑥))
77	70, 76	eqtrid 2790	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = (𝐺‘𝑥))
78	45, 46, 47, 66, 77	seqhomo 13698	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
79	44, 78	eqtr3d 2780	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
80	79	mpteq2dva 5170	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
81	39, 80	eqtrd 2778	1 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509  ℤ_≥cuz 12511  ...cfz 13168  seqcseq 13649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650

This theorem is referenced by:  seqof2  13709  mtest  25468  pserulm  25486  knoppcnlem7  34606