Theorem isercolllem3 15023

Description: Lemma for isercoll 15024. (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		addid2 10823	. . 3 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
2	1	adantl 484	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3		addid1 10820	. . 3 ⊢ (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
4	3	adantl 484	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5		addcl 10619	. . 3 ⊢ ((𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑛 + 𝑘) ∈ ℂ)
6	5	adantl 484	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ (𝑛 ∈ ℂ ∧ 𝑘 ∈ ℂ)) → (𝑛 + 𝑘) ∈ ℂ)
7		0cnd 10634	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 0 ∈ ℂ)
8		cnvimass 5949	. . . . 5 ⊢ (◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺
9		isercoll.g	. . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶𝑍)
10	9	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝐺:ℕ⟶𝑍)
11	8, 10	fssdm 6530	. . . 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 15021	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
16	11, 15	syldan 593	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
17	12, 13, 9, 14	isercolllem2 15022	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
18		isoeq4 7073	. . . 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 259	. 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 4192	. . . . 5 ⊢ ((◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
23	21, 22	sylib 220	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
24		1nn 11649	. . . . . . 7 ⊢ 1 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ ℕ)
26		ffvelrn 6849	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
27	9, 24, 26	sylancl 588	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
28	27, 12	eleqtrdi 2923	. . . . . . . 8 ⊢ (𝜑 → (𝐺‘1) ∈ (ℤ≥‘𝑀))
29	28	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (ℤ≥‘𝑀))
30		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝑁 ∈ (ℤ≥‘(𝐺‘1)))
31		elfzuzb 12903	. . . . . . 7 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))))
32	29, 30, 31	sylanbrc 585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
33		ffn 6514	. . . . . . 7 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
34		elpreima 6828	. . . . . . 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 711	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ (◡𝐺 “ (𝑀...𝑁)))
37	36	ne0d 4301	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
38	23, 37	eqnetrd 3083	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
39		imadisj 5948	. . . 4 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = ∅)
40	39	necon3bii 3068	. . 3 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
41	38, 40	sylibr 236	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
42		ffun 6517	. . . 4 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
43		funimacnv 6435	. . . 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 4205	. . 3 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
46	44, 45	eqsstrdi 4021	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
47		simpl 485	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝜑)
48		elfzuz 12905	. . . 4 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
49	48, 12	eleqtrrdi 2924	. . 3 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ 𝑍)
50		isercoll.f	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)
51	47, 49, 50	syl2an 597	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ ℂ)
52	44	difeq2d 4099	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
53		difin 4238	. . . . . 6 ⊢ ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
54	52, 53	syl6eq 2872	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
55	49	ssriv 3971	. . . . . 6 ⊢ (𝑀...𝑁) ⊆ 𝑍
56		ssdif 4116	. . . . . 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 4005	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
59	58	sselda 3967	. . 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 713	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
62	59, 61	syldan 593	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → (𝐹‘𝑛) = 0)
63		elfznn 12937	. . . 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 597	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
66	17	eleq2d 2898	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) ↔ 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁))))
67	66	biimpa 479	. . . . 5 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → 𝑘 ∈ (◡𝐺 “ (𝑀...𝑁)))
68	67	fvresd 6690	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺‘𝑘))
69	68	fveq2d 6674	. . 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 2859	. 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 13824	1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3016   ∖ cdif 3933   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291   class class class wbr 5066  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   ↾ cres 5557   “ cima 5558  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355   Isom wiso 6356  (class class class)co 7156  ℂcc 10535  0cc0 10537  1c1 10538   + caddc 10540   < clt 10675  ℕcn 11638  ℤcz 11982  ℤ_≥cuz 12244  ...cfz 12893  seqcseq 13370  ♯chash 13691

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-seq 13371  df-hash 13692

This theorem is referenced by:  isercoll  15024