Theorem seqof 13958

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 6830	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
3	2	rgenw 3049	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V
4		eqid 2730	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))
5	4	fnmpt 6617	. . . . . . . 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 6570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) Fn 𝐴 ↔ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴))
9	6, 8	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) Fn 𝐴)
10		fvex 6830	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		fneq1 6568	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹‘𝑥) Fn 𝐴))
12	10, 11	elab 3633	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝐹‘𝑥) Fn 𝐴)
13	9, 12	sylibr 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
14		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15		simprr 772	. . . . . . . . 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 4175	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	14, 15, 17, 17, 18	offn 7618	. . . . . . . 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 3438	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22		fneq1 6568	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴))
23	21, 22	elab 3633	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24		vex 3438	. . . . . . . . 9 ⊢ 𝑦 ∈ V
25		fneq1 6568	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
26	24, 25	elab 3633	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑦 Fn 𝐴)
27	23, 26	anbi12i 628	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) ↔ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴))
28		ovex 7374	. . . . . . . 8 ⊢ (𝑥 ∘f + 𝑦) ∈ V
29		fneq1 6568	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∘f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥 ∘f + 𝑦) Fn 𝐴))
30	28, 29	elab 3633	. . . . . . 7 ⊢ ((𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝑥 ∘f + 𝑦) Fn 𝐴)
31	20, 27, 30	3imtr4g 296	. . . . . 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 13921	. . . 4 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
34		fvex 6830	. . . . 5 ⊢ (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35		fneq1 6568	. . . . 5 ⊢ (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
36	34, 35	elab 3633	. . . 4 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
37	33, 36	sylib 218	. . 3 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
38		dffn5 6875	. . 3 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
39	37, 38	sylib 218	. 2 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
40		fveq1 6816	. . . . . 6 ⊢ (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤‘𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41		eqid 2730	. . . . . 6 ⊢ (𝑤 ∈ V ↦ (𝑤‘𝑧)) = (𝑤 ∈ V ↦ (𝑤‘𝑧))
42		fvex 6830	. . . . . 6 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
43	40, 41, 42	fvmpt 6924	. . . . 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 715	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
46	13	adantlr 715	. . . . 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 2731	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘𝑧) = (𝑥‘𝑧))
49		eqidd 2731	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘𝑧) = (𝑦‘𝑧))
50	14, 15, 17, 17, 18, 48, 49	ofval 7616	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
51	50	an32s 652	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
52		fveq1 6816	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∘f + 𝑦) → (𝑤‘𝑧) = ((𝑥 ∘f + 𝑦)‘𝑧))
53		fvex 6830	. . . . . . . . 9 ⊢ ((𝑥 ∘f + 𝑦)‘𝑧) ∈ V
54	52, 41, 53	fvmpt 6924	. . . . . . . 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 6816	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑧) = (𝑥‘𝑧))
57		fvex 6830	. . . . . . . . . 10 ⊢ (𝑥‘𝑧) ∈ V
58	56, 41, 57	fvmpt 6924	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧))
59	58	elv 3439	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧)
60		fveq1 6816	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤‘𝑧) = (𝑦‘𝑧))
61		fvex 6830	. . . . . . . . . 10 ⊢ (𝑦‘𝑧) ∈ V
62	60, 41, 61	fvmpt 6924	. . . . . . . . 9 ⊢ (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧))
63	62	elv 3439	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧)
64	59, 63	oveq12i 7353	. . . . . . 7 ⊢ (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)) = ((𝑥‘𝑧) + (𝑦‘𝑧))
65	51, 55, 64	3eqtr4g 2790	. . . . . 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 594	. . . . 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 6816	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝑤‘𝑧) = ((𝐹‘𝑥)‘𝑧))
68		fvex 6830	. . . . . . . 8 ⊢ ((𝐹‘𝑥)‘𝑧) ∈ V
69	67, 41, 68	fvmpt 6924	. . . . . . 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 715	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
72	71	fveq1d 6819	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧))
73		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧 ∈ 𝐴)
74	4	fvmpt2 6935	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑥) ∈ V) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
75	73, 2, 74	sylancl 586	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
76	72, 75	eqtrd 2765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = (𝐺‘𝑥))
77	70, 76	eqtrid 2777	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = (𝐺‘𝑥))
78	45, 46, 47, 66, 77	seqhomo 13948	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
79	44, 78	eqtr3d 2767	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
80	79	mpteq2dva 5182	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
81	39, 80	eqtrd 2765	1 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  {cab 2708  ∀wral 3045  Vcvv 3434   ↦ cmpt 5170   Fn wfn 6472  ‘cfv 6477  (class class class)co 7341   ∘_f cof 7603  ℤ_≥cuz 12724  ...cfz 13399  seqcseq 13900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-of 7605  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-n0 12374  df-z 12461  df-uz 12725  df-fz 13400  df-seq 13901

This theorem is referenced by:  seqof2  13959  mtest  26333  pserulm  26351  knoppcnlem7  36512