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Theorem isercoll2 15232

Description: Generalize isercoll 15231 so that both sequences have arbitrary starting point. (Contributed by Mario Carneiro, 6-Apr-2015.)

**Hypotheses**
Ref	Expression
isercoll2.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
isercoll2.w	⊢ 𝑊 = (ℤ_≥‘𝑁)
isercoll2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
isercoll2.n	⊢ (𝜑 → 𝑁 ∈ ℤ)
isercoll2.g	⊢ (𝜑 → 𝐺:𝑍⟶𝑊)
isercoll2.i	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
isercoll2.0	⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
isercoll2.f	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ)
isercoll2.h	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))

**Assertion**
Ref	Expression
isercoll2	⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))

Distinct variable groups: 𝑘,𝑛,𝐴 𝑘,𝐹,𝑛 𝑘,𝐺,𝑛 𝑘,𝐻,𝑛 𝑛,𝑁 𝑘,𝑀,𝑛 𝜑,𝑘,𝑛 𝑛,𝑊 𝑘,𝑍 Allowed substitution hints: 𝑁(𝑘) 𝑊(𝑘) 𝑍(𝑛)

Proof of Theorem isercoll2 Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isercoll2.z	. . 3 ⊢ 𝑍 = (ℤ_≥‘𝑀)
2		isercoll2.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		1z 12207	. . . 4 ⊢ 1 ∈ ℤ
4		zsubcl 12219	. . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 − 𝑀) ∈ ℤ)
5	3, 2, 4	sylancr 590	. . 3 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		seqex 13576	. . . 4 ⊢ seq𝑀( + , 𝐻) ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐻) ∈ V)
8		seqex 13576	. . . 4 ⊢ seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ∈ V
9	8	a1i 11	. . 3 ⊢ (𝜑 → seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ∈ V)
10		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
11	10, 1	eleqtrdi 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
12	5	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 − 𝑀) ∈ ℤ)
13		simpl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝜑)
14		elfzuz 13108	. . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ (ℤ_≥‘𝑀))
15	14, 1	eleqtrrdi 2849	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ 𝑍)
16		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
17	16, 1	eleqtrdi 2848	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
18		eluzelz 12448	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
20	19	zcnd 12283	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℂ)
21	2	zcnd 12283	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
22	21	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ ℂ)
23		1cnd 10828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 1 ∈ ℂ)
24	20, 22, 23	subadd23d 11211	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 − 𝑀) + 1) = (𝑗 + (1 − 𝑀)))
25		uznn0sub 12473	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (𝑗 − 𝑀) ∈ ℕ₀)
26	17, 25	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈ ℕ₀)
27		nn0p1nn 12129	. . . . . . . . . 10 ⊢ ((𝑗 − 𝑀) ∈ ℕ₀ → ((𝑗 − 𝑀) + 1) ∈ ℕ)
28	26, 27	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 − 𝑀) + 1) ∈ ℕ)
29	24, 28	eqeltrrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + (1 − 𝑀)) ∈ ℕ)
30		oveq1 7220	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑥 − 1) = ((𝑗 + (1 − 𝑀)) − 1))
31	30	oveq2d 7229	. . . . . . . . . 10 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)))
32	31	fveq2d 6721	. . . . . . . . 9 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))))
33		eqid 2737	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))
34		fvex 6730	. . . . . . . . 9 ⊢ (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) ∈ V
35	32, 33, 34	fvmpt 6818	. . . . . . . 8 ⊢ ((𝑗 + (1 − 𝑀)) ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))))
36	29, 35	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))))
37	24	oveq1d 7228	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = ((𝑗 + (1 − 𝑀)) − 1))
38	26	nn0cnd 12152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈ ℂ)
39		ax-1cn 10787	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
40		pncan 11084	. . . . . . . . . . . 12 ⊢ (((𝑗 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀))
41	38, 39, 40	sylancl 589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀))
42	37, 41	eqtr3d 2779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 + (1 − 𝑀)) − 1) = (𝑗 − 𝑀))
43	42	oveq2d 7229	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = (𝑀 + (𝑗 − 𝑀)))
44	22, 20	pncan3d 11192	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + (𝑗 − 𝑀)) = 𝑗)
45	43, 44	eqtrd 2777	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = 𝑗)
46	45	fveq2d 6721	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) = (𝐻‘𝑗))
47	36, 46	eqtr2d 2778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))))
48	13, 15, 47	syl2an 599	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑘)) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))))
49	11, 12, 48	seqshft2 13602	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( + , 𝐻)‘𝑘) = (seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))))
50	21	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℂ)
51		pncan3 11086	. . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + (1 − 𝑀)) = 1)
52	50, 39, 51	sylancl 589	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (1 − 𝑀)) = 1)
53	52	seqeq1d 13580	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) = seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))))
54	53	fveq1d 6719	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))) = (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))))
55	49, 54	eqtr2d 2778	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))) = (seq𝑀( + , 𝐻)‘𝑘))
56	1, 2, 5, 7, 9, 55	climshft2 15143	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ⇝ 𝐴))
57		isercoll2.w	. . 3 ⊢ 𝑊 = (ℤ_≥‘𝑁)
58		isercoll2.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
59		isercoll2.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑍⟶𝑊)
60	59	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:𝑍⟶𝑊)
61		uzid 12453	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
62	2, 61	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
63		nnm1nn0 12131	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑥 − 1) ∈ ℕ₀)
64		uzaddcl 12500	. . . . . . 7 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑀) ∧ (𝑥 − 1) ∈ ℕ₀) → (𝑀 + (𝑥 − 1)) ∈ (ℤ_≥‘𝑀))
65	62, 63, 64	syl2an 599	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈ (ℤ_≥‘𝑀))
66	65, 1	eleqtrrdi 2849	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈ 𝑍)
67	60, 66	ffvelrnd 6905	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐺‘(𝑀 + (𝑥 − 1))) ∈ 𝑊)
68	67	fmpttd 6932	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))):ℕ⟶𝑊)
69		fveq2 6717	. . . . . . 7 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑗 − 1))))
70		fvoveq1 7236	. . . . . . 7 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘(𝑘 + 1)) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
71	69, 70	breq12d 5066	. . . . . 6 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → ((𝐺‘𝑘) < (𝐺‘(𝑘 + 1)) ↔ (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘((𝑀 + (𝑗 − 1)) + 1))))
72		isercoll2.i	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
73	72	ralrimiva 3105	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
74	73	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
75		nnm1nn0 12131	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ₀)
76		uzaddcl 12500	. . . . . . . 8 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑀) ∧ (𝑗 − 1) ∈ ℕ₀) → (𝑀 + (𝑗 − 1)) ∈ (ℤ_≥‘𝑀))
77	62, 75, 76	syl2an 599	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈ (ℤ_≥‘𝑀))
78	77, 1	eleqtrrdi 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈ 𝑍)
79	71, 74, 78	rspcdva 3539	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
80		nncn 11838	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
81	80	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
82		1cnd 10828	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
83	81, 82, 82	addsubd 11210	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 + 1) − 1) = ((𝑗 − 1) + 1))
84	83	oveq2d 7229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = (𝑀 + ((𝑗 − 1) + 1)))
85	21	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑀 ∈ ℂ)
86	75	adantl 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈ ℕ₀)
87	86	nn0cnd 12152	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈ ℂ)
88	85, 87, 82	addassd 10855	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑀 + (𝑗 − 1)) + 1) = (𝑀 + ((𝑗 − 1) + 1)))
89	84, 88	eqtr4d 2780	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = ((𝑀 + (𝑗 − 1)) + 1))
90	89	fveq2d 6721	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
91	79, 90	breqtrrd 5081	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
92		oveq1 7220	. . . . . . . 8 ⊢ (𝑥 = 𝑗 → (𝑥 − 1) = (𝑗 − 1))
93	92	oveq2d 7229	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑀 + (𝑥 − 1)) = (𝑀 + (𝑗 − 1)))
94	93	fveq2d 6721	. . . . . 6 ⊢ (𝑥 = 𝑗 → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (𝑗 − 1))))
95		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))
96		fvex 6730	. . . . . 6 ⊢ (𝐺‘(𝑀 + (𝑗 − 1))) ∈ V
97	94, 95, 96	fvmpt 6818	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1))))
98	97	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1))))
99		peano2nn 11842	. . . . . 6 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
100	99	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
101		oveq1 7220	. . . . . . . 8 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 − 1) = ((𝑗 + 1) − 1))
102	101	oveq2d 7229	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + 1) − 1)))
103	102	fveq2d 6721	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
104		fvex 6730	. . . . . 6 ⊢ (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) ∈ V
105	103, 95, 104	fvmpt 6818	. . . . 5 ⊢ ((𝑗 + 1) ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
106	100, 105	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
107	91, 98, 106	3brtr4d 5085	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) < ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)))
108	59	ffnd 6546	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝑍)
109		uznn0sub 12473	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 − 𝑀) ∈ ℕ₀)
110	11, 109	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈ ℕ₀)
111		nn0p1nn 12129	. . . . . . . . . . . 12 ⊢ ((𝑘 − 𝑀) ∈ ℕ₀ → ((𝑘 − 𝑀) + 1) ∈ ℕ)
112	110, 111	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 − 𝑀) + 1) ∈ ℕ)
113	110	nn0cnd 12152	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈ ℂ)
114		pncan 11084	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀))
115	113, 39, 114	sylancl 589	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀))
116	115	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (((𝑘 − 𝑀) + 1) − 1)) = (𝑀 + (𝑘 − 𝑀)))
117		eluzelz 12448	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
118	117, 1	eleq2s 2856	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
119	118	zcnd 12283	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ)
120		pncan3 11086	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑀 + (𝑘 − 𝑀)) = 𝑘)
121	21, 119, 120	syl2an 599	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (𝑘 − 𝑀)) = 𝑘)
122	116, 121	eqtr2d 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 = (𝑀 + (((𝑘 − 𝑀) + 1) − 1)))
123	122	fveq2d 6721	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1))))
124		oveq1 7220	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑥 − 1) = (((𝑘 − 𝑀) + 1) − 1))
125	124	oveq2d 7229	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + (((𝑘 − 𝑀) + 1) − 1)))
126	125	fveq2d 6721	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1))))
127	126	rspceeqv 3552	. . . . . . . . . . 11 ⊢ ((((𝑘 − 𝑀) + 1) ∈ ℕ ∧ (𝐺‘𝑘) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1)))) → ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1))))
128	112, 123, 127	syl2anc 587	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1))))
129		fvex 6730	. . . . . . . . . . 11 ⊢ (𝐺‘𝑘) ∈ V
130	95	elrnmpt 5825	. . . . . . . . . . 11 ⊢ ((𝐺‘𝑘) ∈ V → ((𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1)))))
131	129, 130	ax-mp 5	. . . . . . . . . 10 ⊢ ((𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1))))
132	128, 131	sylibr 237	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
133	132	ralrimiva 3105	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
134		ffnfv 6935	. . . . . . . 8 ⊢ (𝐺:𝑍⟶ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) ↔ (𝐺 Fn 𝑍 ∧ ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))))
135	108, 133, 134	sylanbrc 586	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
136	135	frnd 6553	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
137	136	sscond 4056	. . . . 5 ⊢ (𝜑 → (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) ⊆ (𝑊 ∖ ran 𝐺))
138	137	sselda 3901	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))) → 𝑛 ∈ (𝑊 ∖ ran 𝐺))
139		isercoll2.0	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran 𝐺)) → (𝐹‘𝑛) = 0)
140	138, 139	syldan 594	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))) → (𝐹‘𝑛) = 0)
141		isercoll2.f	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑊) → (𝐹‘𝑛) ∈ ℂ)
142		fveq2 6717	. . . . . 6 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐻‘𝑘) = (𝐻‘(𝑀 + (𝑗 − 1))))
143	69	fveq2d 6721	. . . . . 6 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
144	142, 143	eqeq12d 2753	. . . . 5 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → ((𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)) ↔ (𝐻‘(𝑀 + (𝑗 − 1))) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1))))))
145		isercoll2.h	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
146	145	ralrimiva 3105	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
147	146	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
148	144, 147, 78	rspcdva 3539	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑀 + (𝑗 − 1))) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
149	93	fveq2d 6721	. . . . . 6 ⊢ (𝑥 = 𝑗 → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + (𝑗 − 1))))
150		fvex 6730	. . . . . 6 ⊢ (𝐻‘(𝑀 + (𝑗 − 1))) ∈ V
151	149, 33, 150	fvmpt 6818	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1))))
152	151	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1))))
153	98	fveq2d 6721	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
154	148, 152, 153	3eqtr4d 2787	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐹‘((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗)))
155	57, 58, 68, 107, 140, 141, 154	isercoll 15231	. 2 ⊢ (𝜑 → (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))
156	56, 155	bitrd 282	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ∖ cdif 3863 class class class wbr 5053 ↦ cmpt 5135 ran crn 5552 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 1c1 10730 + caddc 10732 < clt 10867 − cmin 11062 ℕcn 11830 ℕ₀cn0 12090 ℤcz 12176 ℤ_≥cuz 12438 ...cfz 13095 seqcseq 13574 ⇝ cli 15045

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-oadd 8206 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-n0 12091 df-xnn0 12163 df-z 12177 df-uz 12439 df-fz 13096 df-seq 13575 df-hash 13897 df-shft 14630 df-clim 15049

This theorem is referenced by: iserodd 16388 stirlinglem5 43294

W3C validator