Theorem isercolllem3 15875

Description: Lemma for isercoll 15876. (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 11530	. . 3 ⊢ (𝑛 ∈ ℂ → (0 + 𝑛) = 𝑛)
2	1	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (0 + 𝑛) = 𝑛)
3		addrid 11527	. . 3 ⊢ (𝑛 ∈ ℂ → (𝑛 + 0) = 𝑛)
4	3	adantl 487	. 2 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑛 ∈ ℂ) → (𝑛 + 0) = 𝑛)
5		addcl 11319	. . 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 11336	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 0 ∈ ℂ)
8		cnvimass 6215	. . . . 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 6757	. . . 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 15873	. . . 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 15874	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁))))) = (◡𝐺 “ (𝑀...𝑁)))
18		isoeq4 7357	. . . 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 4123	. . . . 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 12384	. . . . . . 7 ⊢ 1 ∈ ℕ
25	24	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 1 ∈ ℕ)
26		ffvelcdm 7111	. . . . . . . . . 10 ⊢ ((𝐺:ℕ⟶𝑍 ∧ 1 ∈ ℕ) → (𝐺‘1) ∈ 𝑍)
27	9, 24, 26	sylancl 598	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) ∈ 𝑍)
28	27, 12	eleqtrdi 2874	. . . . . . . 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 13688	. . . . . . 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 6737	. . . . . . 7 ⊢ (𝐺:ℕ⟶𝑍 → 𝐺 Fn ℕ)
34		elpreima 7086	. . . . . . 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 4291	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (◡𝐺 “ (𝑀...𝑁)) ≠ ∅)
38	23, 37	eqnetrd 3026	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) ≠ ∅)
39		imadisj 6093	. . . 4 ⊢ ((𝐺 “ (◡𝐺 “ (𝑀...𝑁))) = ∅ ↔ (dom 𝐺 ∩ (◡𝐺 “ (𝑀...𝑁))) = ∅)
40	39	necon3bii 3011	. . 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 6740	. . . 4 ⊢ (𝐺:ℕ⟶𝑍 → Fun 𝐺)
43		funimacnv 6649	. . . 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 4136	. . 3 ⊢ ((𝑀...𝑁) ∩ ran 𝐺) ⊆ (𝑀...𝑁)
46	44, 45	eqsstrdi 3957	. 2 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → (𝐺 “ (◡𝐺 “ (𝑀...𝑁))) ⊆ (𝑀...𝑁))
47		simpl 488	. . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → 𝜑)
48		elfzuz 13690	. . . 4 ⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀))
49	48, 12	eleqtrrdi 2875	. . 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 4066	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)))
53		difin 4220	. . . . . 6 ⊢ ((𝑀...𝑁) ∖ ((𝑀...𝑁) ∩ ran 𝐺)) = ((𝑀...𝑁) ∖ ran 𝐺)
54	52, 53	eqtrdi 2814	. . . . 5 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) = ((𝑀...𝑁) ∖ ran 𝐺))
55	49	ssriv 3917	. . . . . 6 ⊢ (𝑀...𝑁) ⊆ 𝑍
56		ssdif 4083	. . . . . 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 3947	. . . 4 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) → ((𝑀...𝑁) ∖ (𝐺 “ (◡𝐺 “ (𝑀...𝑁)))) ⊆ (𝑍 ∖ ran 𝐺))
59	58	sselda 3913	. . 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 13725	. . . 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 2850	. . . . . 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 6934	. . . 4 ⊢ (((𝜑 ∧ 𝑁 ∈ (ℤ≥‘(𝐺‘1))) ∧ 𝑘 ∈ (1...(♯‘(𝐺 “ (◡𝐺 “ (𝑀...𝑁)))))) → ((𝐺 ↾ (◡𝐺 “ (𝑀...𝑁)))‘𝑘) = (𝐺‘𝑘))
69	68	fveq2d 6918	. . 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 2801	. 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 14649	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 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281   class class class wbr 5106  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   ↾ cres 5658   “ cima 5659  Fun wfun 6558   Fn wfn 6559  ⟶wf 6560  ‘cfv 6564   Isom wiso 6565  (class class class)co 7449  ℂcc 11235  0cc0 11237  1c1 11238   + caddc 11240   < clt 11380  ℕcn 12373  ℤcz 12731  ℤ_≥cuz 13003  ...cfz 13677  seqcseq 14182  ♯chash 14513

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-isom 6573  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-er 8753  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-card 10055  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-n0 12645  df-z 12732  df-uz 13004  df-fz 13678  df-seq 14183  df-hash 14514

This theorem is used by:  isercoll  15876