Theorem isercolllem3 15378

Description: Lemma for isercoll 15379. (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 11158	. . 3 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
2	1	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3		addid1 11155	. . 3 ⊢ (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
4	3	adantl 482	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5		addcl 10953	. . 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 10968	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 0 ∈ ℂ)
8		cnvimass 5989	. . . . 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 6620	. . . 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 15376	. . . 4 ⊢ ((𝜑 ∧ (◡𝐺 “ (𝑀...𝑁)) ⊆ ℕ) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
16	11, 15	syldan 591	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 ↾ (◡𝐺 “ (𝑀...𝑁))) Isom < , < ((◡𝐺 “ (𝑀...𝑁)), (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))
17	12, 13, 9, 14	isercolllem2 15377	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
18		isoeq4 7191	. . . 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 256	. 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 4149	. . . . 5 ⊢ ((◡𝐺 “ (𝑀...𝑁)) ⊆ dom 𝐺 ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
23	21, 22	sylib 217	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = (◡𝐺 “ (𝑀...𝑁)))
24		1nn 11984	. . . . . . 7 ⊢ 1 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ ℕ)
26		ffvelrn 6959	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
27	9, 24, 26	sylancl 586	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
28	27, 12	eleqtrdi 2849	. . . . . . . 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 13250	. . . . . . 7 ⊢ ((𝐺‘1) ∈ (𝑀...𝑁) ↔ ((𝐺‘1) ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))))
32	29, 30, 31	sylanbrc 583	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺‘1) ∈ (𝑀...𝑁))
33		ffn 6600	. . . . . . 7 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
34		elpreima 6935	. . . . . . 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 710	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ (◡𝐺 “ (𝑀...𝑁)))
37	36	ne0d 4269	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
38	23, 37	eqnetrd 3011	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
39		imadisj 5988	. . . 4 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = ∅)
40	39	necon3bii 2996	. . 3 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
41	38, 40	sylibr 233	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
42		ffun 6603	. . . 4 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
43		funimacnv 6515	. . . 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 4162	. . 3 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
46	44, 45	eqsstrdi 3975	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
47		simpl 483	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝜑)
48		elfzuz 13252	. . . 4 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
49	48, 12	eleqtrrdi 2850	. . 3 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ 𝑍)
50		isercoll.f	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ ℂ)
51	47, 49, 50	syl2an 596	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑀...𝑁)) → (𝐹‘𝑛) ∈ ℂ)
52	44	difeq2d 4057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
53		difin 4195	. . . . . 6 ⊢ ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
54	52, 53	eqtrdi 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
55	49	ssriv 3925	. . . . . 6 ⊢ (𝑀...𝑁) ⊆ 𝑍
56		ssdif 4074	. . . . . 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 3959	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
59	58	sselda 3921	. . 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 712	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ (𝑍 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
62	59, 61	syldan 591	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) → (𝐹‘𝑛) = 0)
63		elfznn 13285	. . . 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 596	. . 3 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
66	17	eleq2d 2824	. . . . . 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 6794	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺‘𝑘))
69	68	fveq2d 6778	. . 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 2781	. 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 14179	1 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (seq𝑀( + , 𝐹)‘𝑁) = (seq1( + , 𝐻)‘(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∖ cdif 3884   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074  ^◡ccnv 5588  dom cdm 5589  ran crn 5590   ↾ cres 5591   “ cima 5592  Fun wfun 6427   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433   Isom wiso 6434  (class class class)co 7275  ℂcc 10869  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009  ℕcn 11973  ℤcz 12319  ℤ_≥cuz 12582  ...cfz 13239  seqcseq 13721  ♯chash 14044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-seq 13722  df-hash 14045

This theorem is referenced by:  isercoll  15379