Theorem isercolllem3 15627

Description: Lemma for isercoll 15628. (Contributed by Mario Carneiro, 6-Apr-2015.)

Hypotheses

Ref	Expression
isercoll.z	⊢ 𝑍 = (ℤ≥‘𝑀)
isercoll.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
isercoll.g	⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
isercoll.i	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
isercoll.0	⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
isercoll.f	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)
isercoll.h	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))

Assertion

Ref	Expression
isercolllem3	⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))

Distinct variable groups:   𝑘,𝑛,𝐹   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝑘,𝐺,𝑛   𝑘,𝐻,𝑛   𝑘,𝑀,𝑛   𝑛,𝑍

Allowed substitution hint:   𝑍(𝑘)

Proof of Theorem isercolllem3

Step	Hyp	Ref	Expression
1		addlid 11327	. . 3 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
2	1	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3		addrid 11324	. . 3 ⊢ (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
4	3	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5		addcl 11118	. . 3 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑛 + 𝑘) ∈ ℂ)
6	5	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ (𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ)) → (𝑛 + 𝑘) ∈ ℂ)
7		0cnd 11135	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 0 ∈ ℂ)
8		cnvimass 6041	. . . . 5 ⊢ (◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
9		isercoll.g	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
10	9	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
11	8, 10	fssdm 6681	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ)
12		isercoll.z	. . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀)
13		isercoll.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
14		isercoll.i	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
15	12, 13, 9, 14	isercolllem1 15625	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
16	11, 15	syldan 597	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
17	12, 13, 9, 14	isercolllem2 15626	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
18		isoeq4 7271	. . . 4 ⊢ ((1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))
19	17, 18	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ↔ (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))
20	16, 19	mpbird 258	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
21	8	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺)
22		sseqin2 4159	. . . . 5 ⊢ ((◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
23	21, 22	sylib 219	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
24		1nn 12183	. . . . . . 7 ⊢ 1 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ ℕ)
26		ffvelcdm 7029	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
27	9, 24, 26	sylancl 592	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
28	27, 12	eleqtrdi 2850	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ≥‘𝑀))
29	28	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ≥‘𝑀))
30		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ≥‘(𝐺‘1)))
31		elfzuzb 13470	. . . . . . 7 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))))
32	29, 30, 31	sylanbrc 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
33		ffn 6662	. . . . . . 7 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
34		elpreima 7006	. . . . . . 7 ⊢ (𝐺 Fn ℕ → (1 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
35	10, 33, 34	3syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
36	25, 32, 35	mpbir2and 719	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ (◡𝐺 “ (𝑀...𝑁)))
37	36	ne0d 4277	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
38	23, 37	eqnetrd 3002	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
39		imadisj 6039	. . . 4 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = ∅)
40	39	necon3bii 2987	. . 3 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
41	38, 40	sylibr 235	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
42		ffun 6665	. . . 4 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
43		funimacnv 6573	. . . 4 ⊢ (Fun 𝐺 → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
44	10, 42, 43	3syl 18	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
45		inss1 4172	. . 3 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
46	44, 45	eqsstrdi 3966	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
47		simpl 483	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝜑)
48		elfzuz 13472	. . . 4 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
49	48, 12	eleqtrrdi 2851	. . 3 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ 𝑍)
50		isercoll.f	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)
51	47, 49, 50	syl2an 602	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ ℂ)
52	44	difeq2d 4064	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
53		difin 4207	. . . . . 6 ⊢ ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
54	52, 53	eqtrdi 2791	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
55	49	ssriv 3926	. . . . . 6 ⊢ (𝑀...𝑁) ⊆ 𝑍
56		ssdif 4081	. . . . . 6 ⊢ ((𝑀...𝑁) ⊆ 𝑍 → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
57	55, 56	mp1i 13	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
58	54, 57	eqsstrd 3956	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
59	58	sselda 3922	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → 𝑛 ∈ (𝑍 ∖ ran 𝐺))
60		isercoll.0	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
61	60	adantlr 721	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
62	59, 61	syldan 597	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → (𝐹‘𝑛) = 0)
63		elfznn 13505	. . . 4 ⊢ (𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → 𝑘 ∈ ℕ)
64		isercoll.h	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
65	47, 63, 64	syl2an 602	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
66	17	eleq2d 2826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) ↔ 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁))))
67	66	biimpa 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁)))
68	67	fvresd 6854	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺‘𝑘))
69	68	fveq2d 6838	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐹‘((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘)) = (𝐹‘(𝐺‘𝑘)))
70	65, 69	eqtr4d 2778	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐻‘𝑘) = (𝐹‘((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘)))
71	2, 4, 6, 7, 20, 41, 46, 51, 62, 70	seqcoll2 14425	1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   ∖ cdif 3887   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268   class class class wbr 5079  ^◡ccnv 5624  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Fun wfun 6486   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492   Isom wiso 6493  (class class class)co 7363  ℂcc 11034  0cc0 11036  1c1 11037   + caddc 11039   < clt 11177  ℕcn 12172  ℤcz 12522  ℤ_≥cuz 12786  ...cfz 13459  seqcseq 13961  ♯chash 14290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113  ax-pre-sup 11114

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-sup 9352  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-seq 13962  df-hash 14291

This theorem is referenced by:  isercoll  15628