Theorem isercolllem3 15840

Description: Lemma for isercoll 15841. (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 11497	. . 3 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
2	1	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3		addrid 11494	. . 3 ⊢ (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
4	3	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5		addcl 11286	. . 3 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑛 + 𝑘) ∈ ℂ)
6	5	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ (𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ)) → (𝑛 + 𝑘) ∈ ℂ)
7		0cnd 11303	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 0 ∈ ℂ)
8		cnvimass 6198	. . . . 5 ⊢ (◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
9		isercoll.g	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
10	9	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
11	8, 10	fssdm 6730	. . . 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 15838	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
16	11, 15	syldan 603	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
17	12, 13, 9, 14	isercolllem2 15839	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
18		isoeq4 7329	. . . 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 18	. . 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 260	. 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 4120	. . . . 5 ⊢ ((◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
23	21, 22	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
24		1nn 12350	. . . . . . 7 ⊢ 1 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ ℕ)
26		ffvelcdm 7082	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
27	9, 24, 26	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
28	27, 12	eleqtrdi 2870	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ≥‘𝑀))
29	28	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ≥‘𝑀))
30		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ≥‘(𝐺‘1)))
31		elfzuzb 13654	. . . . . . 7 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))))
32	29, 30, 31	sylanbrc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
33		ffn 6710	. . . . . . 7 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
34		elpreima 7058	. . . . . . 7 ⊢ (𝐺 Fn ℕ → (1 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
35	10, 33, 34	3syl 19	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1 ∈ (◡𝐺 “ (𝑀...𝑁)) ↔ (1 ∈ ℕ ∧ (𝐺‘1) ∈ (𝑀...𝑁))))
36	25, 32, 35	mpbir2and 726	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ (◡𝐺 “ (𝑀...𝑁)))
37	36	ne0d 4287	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
38	23, 37	eqnetrd 3022	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
39		imadisj 6076	. . . 4 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = ∅)
40	39	necon3bii 3007	. . 3 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
41	38, 40	sylibr 237	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
42		ffun 6713	. . . 4 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
43		funimacnv 6622	. . . 4 ⊢ (Fun 𝐺 → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
44	10, 42, 43	3syl 19	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ((𝑀...𝑁) ∩ ran 𝐺))
45		inss1 4133	. . 3 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
46	44, 45	eqsstrdi 3952	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
47		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝜑)
48		elfzuz 13656	. . . 4 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
49	48, 12	eleqtrrdi 2871	. . 3 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ 𝑍)
50		isercoll.f	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)
51	47, 49, 50	syl2an 608	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ ℂ)
52	44	difeq2d 4063	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
53		difin 4217	. . . . . 6 ⊢ ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
54	52, 53	eqtrdi 2811	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
55	49	ssriv 3912	. . . . . 6 ⊢ (𝑀...𝑁) ⊆ 𝑍
56		ssdif 4080	. . . . . 6 ⊢ ((𝑀...𝑁) ⊆ 𝑍 → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
57	55, 56	mp1i 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ ran 𝐺) ⊆ (𝑍 ∖ ran 𝐺))
58	54, 57	eqsstrd 3942	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
59	58	sselda 3908	. . 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 728	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
62	59, 61	syldan 603	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → (𝐹‘𝑛) = 0)
63		elfznn 13691	. . . 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 608	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
66	17	eleq2d 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) ↔ 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁))))
67	66	biimpa 482	. . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁)))
68	67	fvresd 6906	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺‘𝑘))
69	68	fveq2d 6890	. . 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 2798	. 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 14614	1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278   class class class wbr 5102  ^◡ccnv 5649  dom cdm 5650  ran crn 5651   ↾ cres 5652   “ cima 5653  Fun wfun 6533   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539   Isom wiso 6540  (class class class)co 7421  ℂcc 11202  0cc0 11204  1c1 11205   + caddc 11207   < clt 11347  ℕcn 12339  ℤcz 12697  ℤ_≥cuz 12969  ...cfz 13643  seqcseq 14148  ♯chash 14478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752  ax-cnex 11260  ax-resscn 11261  ax-1cn 11262  ax-icn 11263  ax-addcl 11264  ax-addrcl 11265  ax-mulcl 11266  ax-mulrcl 11267  ax-mulcom 11268  ax-addass 11269  ax-mulass 11270  ax-distr 11271  ax-i2m1 11272  ax-1ne0 11273  ax-1rid 11274  ax-rnegex 11275  ax-rrecex 11276  ax-cnre 11277  ax-pre-lttri 11278  ax-pre-lttrn 11279  ax-pre-ltadd 11280  ax-pre-mulgt0 11281  ax-pre-sup 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-er 8721  df-en 8978  df-dom 8979  df-sdom 8980  df-fin 8981  df-sup 9438  df-card 10024  df-pnf 11349  df-mnf 11350  df-xr 11351  df-ltxr 11352  df-le 11353  df-sub 11547  df-neg 11548  df-nn 12340  df-n0 12611  df-z 12698  df-uz 12970  df-fz 13644  df-seq 14149  df-hash 14479

This theorem is used by:  isercoll  15841