Theorem seqof 13421

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 6677	. . . . . . . . 9 ⊢ (𝐺‘𝑥) ∈ V
3	2	rgenw 3150	. . . . . . . 8 ⊢ ∀𝑧 ∈ 𝐴 (𝐺‘𝑥) ∈ V
4		eqid 2821	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))
5	4	fnmpt 6482	. . . . . . . 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 6440	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥) Fn 𝐴 ↔ (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)) Fn 𝐴))
9	6, 8	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) Fn 𝐴)
10		fvex 6677	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
11		fneq1 6438	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑥) → (𝑧 Fn 𝐴 ↔ (𝐹‘𝑥) Fn 𝐴))
12	10, 11	elab 3666	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝐹‘𝑥) Fn 𝐴)
13	9, 12	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
14		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑥 Fn 𝐴)
15		simprr 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝑦 Fn 𝐴)
16		seqof.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → 𝐴 ∈ 𝑉)
18		inidm 4194	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐴) = 𝐴
19	14, 15, 17, 17, 18	offn 7414	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → (𝑥 ∘f + 𝑦) Fn 𝐴)
20	19	ex 415	. . . . . . 7 ⊢ (𝜑 → ((𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴) → (𝑥 ∘f + 𝑦) Fn 𝐴))
21		vex 3497	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22		fneq1 6438	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 Fn 𝐴 ↔ 𝑥 Fn 𝐴))
23	21, 22	elab 3666	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ 𝑥 Fn 𝐴)
24		vex 3497	. . . . . . . . 9 ⊢ 𝑦 ∈ V
25		fneq1 6438	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑧 Fn 𝐴 ↔ 𝑦 Fn 𝐴))
26	24, 25	elab 3666	. . . . . . . 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 7183	. . . . . . . 8 ⊢ (𝑥 ∘f + 𝑦) ∈ V
29		fneq1 6438	. . . . . . . 8 ⊢ (𝑧 = (𝑥 ∘f + 𝑦) → (𝑧 Fn 𝐴 ↔ (𝑥 ∘f + 𝑦) Fn 𝐴))
30	28, 29	elab 3666	. . . . . . 7 ⊢ ((𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (𝑥 ∘f + 𝑦) Fn 𝐴)
31	20, 27, 30	3imtr4g 298	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴}) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴}))
32	31	imp 409	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
33	1, 13, 32	seqcl 13384	. . . 4 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
34		fvex 6677	. . . . 5 ⊢ (seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ V
35		fneq1 6438	. . . . 5 ⊢ (𝑧 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑧 Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴))
36	34, 35	elab 3666	. . . 4 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
37	33, 36	sylib 220	. . 3 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴)
38		dffn5 6718	. . 3 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁) Fn 𝐴 ↔ (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
39	37, 38	sylib 220	. 2 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)))
40		fveq1 6663	. . . . . 6 ⊢ (𝑤 = (seq𝑀( ∘f + , 𝐹)‘𝑁) → (𝑤‘𝑧) = ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧))
41		eqid 2821	. . . . . 6 ⊢ (𝑤 ∈ V ↦ (𝑤‘𝑧)) = (𝑤 ∈ V ↦ (𝑤‘𝑧))
42		fvex 6677	. . . . . 6 ⊢ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) ∈ V
43	40, 41, 42	fvmpt 6762	. . . . 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 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 ∈ {𝑧 ∣ 𝑧 Fn 𝐴} ∧ 𝑦 ∈ {𝑧 ∣ 𝑧 Fn 𝐴})) → (𝑥 ∘f + 𝑦) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
46	13	adantlr 713	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) ∈ {𝑧 ∣ 𝑧 Fn 𝐴})
47	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑁 ∈ (ℤ≥‘𝑀))
48		eqidd 2822	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥‘𝑧) = (𝑥‘𝑧))
49		eqidd 2822	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑦‘𝑧) = (𝑦‘𝑧))
50	14, 15, 17, 17, 18, 48, 49	ofval 7412	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) ∧ 𝑧 ∈ 𝐴) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
51	50	an32s 650	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ (𝑥 Fn 𝐴 ∧ 𝑦 Fn 𝐴)) → ((𝑥 ∘f + 𝑦)‘𝑧) = ((𝑥‘𝑧) + (𝑦‘𝑧)))
52		fveq1 6663	. . . . . . . . 9 ⊢ (𝑤 = (𝑥 ∘f + 𝑦) → (𝑤‘𝑧) = ((𝑥 ∘f + 𝑦)‘𝑧))
53		fvex 6677	. . . . . . . . 9 ⊢ ((𝑥 ∘f + 𝑦)‘𝑧) ∈ V
54	52, 41, 53	fvmpt 6762	. . . . . . . 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 6663	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑧) = (𝑥‘𝑧))
57		fvex 6677	. . . . . . . . . 10 ⊢ (𝑥‘𝑧) ∈ V
58	56, 41, 57	fvmpt 6762	. . . . . . . . 9 ⊢ (𝑥 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧))
59	58	elv 3499	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) = (𝑥‘𝑧)
60		fveq1 6663	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑤‘𝑧) = (𝑦‘𝑧))
61		fvex 6677	. . . . . . . . . 10 ⊢ (𝑦‘𝑧) ∈ V
62	60, 41, 61	fvmpt 6762	. . . . . . . . 9 ⊢ (𝑦 ∈ V → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧))
63	62	elv 3499	. . . . . . . 8 ⊢ ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦) = (𝑦‘𝑧)
64	59, 63	oveq12i 7162	. . . . . . 7 ⊢ (((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑥) + ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘𝑦)) = ((𝑥‘𝑧) + (𝑦‘𝑧))
65	51, 55, 64	3eqtr4g 2881	. . . . . 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 595	. . . . 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 6663	. . . . . . . 8 ⊢ (𝑤 = (𝐹‘𝑥) → (𝑤‘𝑧) = ((𝐹‘𝑥)‘𝑧))
68		fvex 6677	. . . . . . . 8 ⊢ ((𝐹‘𝑥)‘𝑧) ∈ V
69	67, 41, 68	fvmpt 6762	. . . . . . 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 713	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝐹‘𝑥) = (𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥)))
72	71	fveq1d 6666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧))
73		simplr 767	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑧 ∈ 𝐴)
74	4	fvmpt2 6773	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐴 ∧ (𝐺‘𝑥) ∈ V) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
75	73, 2, 74	sylancl 588	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑧 ∈ 𝐴 ↦ (𝐺‘𝑥))‘𝑧) = (𝐺‘𝑥))
76	72, 75	eqtrd 2856	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝐹‘𝑥)‘𝑧) = (𝐺‘𝑥))
77	70, 76	syl5eq 2868	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ (𝑀...𝑁)) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(𝐹‘𝑥)) = (𝐺‘𝑥))
78	45, 46, 47, 66, 77	seqhomo 13411	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((𝑤 ∈ V ↦ (𝑤‘𝑧))‘(seq𝑀( ∘f + , 𝐹)‘𝑁)) = (seq𝑀( + , 𝐺)‘𝑁))
79	44, 78	eqtr3d 2858	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧) = (seq𝑀( + , 𝐺)‘𝑁))
80	79	mpteq2dva 5153	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐴 ↦ ((seq𝑀( ∘f + , 𝐹)‘𝑁)‘𝑧)) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))
81	39, 80	eqtrd 2856	1 ⊢ (𝜑 → (seq𝑀( ∘f + , 𝐹)‘𝑁) = (𝑧 ∈ 𝐴 ↦ (seq𝑀( + , 𝐺)‘𝑁)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  {cab 2799  ∀wral 3138  Vcvv 3494   ↦ cmpt 5138   Fn wfn 6344  ‘cfv 6349  (class class class)co 7150   ∘_f cof 7401  ℤ_≥cuz 12237  ...cfz 12886  seqcseq 13363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-seq 13364

This theorem is referenced by:  seqof2  13422  mtest  24986  pserulm  25004  knoppcnlem7  33833