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Theorem isercoll2 15573

Description: Generalize isercoll 15572 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 12499	. . . 4 ⊢ 1 ∈ ℤ
4		zsubcl 12511	. . . 4 ⊢ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (1 − 𝑀) ∈ ℤ)
5	3, 2, 4	sylancr 587	. . 3 ⊢ (𝜑 → (1 − 𝑀) ∈ ℤ)
6		seqex 13907	. . . 4 ⊢ seq𝑀( + , 𝐻) ∈ V
7	6	a1i 11	. . 3 ⊢ (𝜑 → seq𝑀( + , 𝐻) ∈ V)
8		seqex 13907	. . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍)
11	10, 1	eleqtrdi 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ_≥‘𝑀))
12	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (1 − 𝑀) ∈ ℤ)
13		simpl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝜑)
14		elfzuz 13417	. . . . . . 7 ⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ (ℤ_≥‘𝑀))
15	14, 1	eleqtrrdi 2842	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...𝑘) → 𝑗 ∈ 𝑍)
16		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍)
17	16, 1	eleqtrdi 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ_≥‘𝑀))
18		eluzelz 12739	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → 𝑗 ∈ ℤ)
19	17, 18	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℤ)
20	19	zcnd 12575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ ℂ)
21	2	zcnd 12575	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
22	21	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑀 ∈ ℂ)
23		1cnd 11104	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 1 ∈ ℂ)
24	20, 22, 23	subadd23d 11491	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 − 𝑀) + 1) = (𝑗 + (1 − 𝑀)))
25		uznn0sub 12768	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑀) → (𝑗 − 𝑀) ∈ ℕ₀)
26	17, 25	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈ ℕ₀)
27		nn0p1nn 12417	. . . . . . . . . 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 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 + (1 − 𝑀)) ∈ ℕ)
30		oveq1 7353	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑥 − 1) = ((𝑗 + (1 − 𝑀)) − 1))
31	30	oveq2d 7362	. . . . . . . . . 10 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)))
32	31	fveq2d 6826	. . . . . . . . 9 ⊢ (𝑥 = (𝑗 + (1 − 𝑀)) → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))))
33		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))
34		fvex 6835	. . . . . . . . 9 ⊢ (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) ∈ V
35	32, 33, 34	fvmpt 6929	. . . . . . . 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 7361	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = ((𝑗 + (1 − 𝑀)) − 1))
38	26	nn0cnd 12441	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑗 − 𝑀) ∈ ℂ)
39		ax-1cn 11061	. . . . . . . . . . . 12 ⊢ 1 ∈ ℂ
40		pncan 11363	. . . . . . . . . . . 12 ⊢ (((𝑗 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀))
41	38, 39, 40	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (((𝑗 − 𝑀) + 1) − 1) = (𝑗 − 𝑀))
42	37, 41	eqtr3d 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ((𝑗 + (1 − 𝑀)) − 1) = (𝑗 − 𝑀))
43	42	oveq2d 7362	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = (𝑀 + (𝑗 − 𝑀)))
44	22, 20	pncan3d 11472	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + (𝑗 − 𝑀)) = 𝑗)
45	43, 44	eqtrd 2766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝑀 + ((𝑗 + (1 − 𝑀)) − 1)) = 𝑗)
46	45	fveq2d 6826	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘(𝑀 + ((𝑗 + (1 − 𝑀)) − 1))) = (𝐻‘𝑗))
47	36, 46	eqtr2d 2767	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))))
48	13, 15, 47	syl2an 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝑍) ∧ 𝑗 ∈ (𝑀...𝑘)) → (𝐻‘𝑗) = ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘(𝑗 + (1 − 𝑀))))
49	11, 12, 48	seqshft2 13932	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (seq𝑀( + , 𝐻)‘𝑘) = (seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1)))))‘(𝑘 + (1 − 𝑀))))
50	21	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑀 ∈ ℂ)
51		pncan3 11365	. . . . . . 7 ⊢ ((𝑀 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑀 + (1 − 𝑀)) = 1)
52	50, 39, 51	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (1 − 𝑀)) = 1)
53	52	seqeq1d 13911	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → seq(𝑀 + (1 − 𝑀))( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) = seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))))
54	53	fveq1d 6824	. . . 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 2767	. . 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 15486	. 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 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → 𝐺:𝑍⟶𝑊)
61		uzid 12744	. . . . . . . 8 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ_≥‘𝑀))
62	2, 61	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘𝑀))
63		nnm1nn0 12419	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → (𝑥 − 1) ∈ ℕ₀)
64		uzaddcl 12799	. . . . . . 7 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑀) ∧ (𝑥 − 1) ∈ ℕ₀) → (𝑀 + (𝑥 − 1)) ∈ (ℤ_≥‘𝑀))
65	62, 63, 64	syl2an 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈ (ℤ_≥‘𝑀))
66	65, 1	eleqtrrdi 2842	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝑀 + (𝑥 − 1)) ∈ 𝑍)
67	60, 66	ffvelcdmd 7018	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐺‘(𝑀 + (𝑥 − 1))) ∈ 𝑊)
68	67	fmpttd 7048	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))):ℕ⟶𝑊)
69		fveq2 6822	. . . . . . 7 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑗 − 1))))
70		fvoveq1 7369	. . . . . . 7 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐺‘(𝑘 + 1)) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
71	69, 70	breq12d 5104	. . . . . 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 3124	. . . . . . 7 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
74	73	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) < (𝐺‘(𝑘 + 1)))
75		nnm1nn0 12419	. . . . . . . 8 ⊢ (𝑗 ∈ ℕ → (𝑗 − 1) ∈ ℕ₀)
76		uzaddcl 12799	. . . . . . . 8 ⊢ ((𝑀 ∈ (ℤ_≥‘𝑀) ∧ (𝑗 − 1) ∈ ℕ₀) → (𝑀 + (𝑗 − 1)) ∈ (ℤ_≥‘𝑀))
77	62, 75, 76	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈ (ℤ_≥‘𝑀))
78	77, 1	eleqtrrdi 2842	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + (𝑗 − 1)) ∈ 𝑍)
79	71, 74, 78	rspcdva 3578	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
80		nncn 12130	. . . . . . . . . 10 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℂ)
81	80	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℂ)
82		1cnd 11104	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 1 ∈ ℂ)
83	81, 82, 82	addsubd 11490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑗 + 1) − 1) = ((𝑗 − 1) + 1))
84	83	oveq2d 7362	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = (𝑀 + ((𝑗 − 1) + 1)))
85	21	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑀 ∈ ℂ)
86	75	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈ ℕ₀)
87	86	nn0cnd 12441	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 − 1) ∈ ℂ)
88	85, 87, 82	addassd 11131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑀 + (𝑗 − 1)) + 1) = (𝑀 + ((𝑗 − 1) + 1)))
89	84, 88	eqtr4d 2769	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑀 + ((𝑗 + 1) − 1)) = ((𝑀 + (𝑗 − 1)) + 1))
90	89	fveq2d 6826	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) = (𝐺‘((𝑀 + (𝑗 − 1)) + 1)))
91	79, 90	breqtrrd 5119	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐺‘(𝑀 + (𝑗 − 1))) < (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
92		oveq1 7353	. . . . . . . 8 ⊢ (𝑥 = 𝑗 → (𝑥 − 1) = (𝑗 − 1))
93	92	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = 𝑗 → (𝑀 + (𝑥 − 1)) = (𝑀 + (𝑗 − 1)))
94	93	fveq2d 6826	. . . . . 6 ⊢ (𝑥 = 𝑗 → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (𝑗 − 1))))
95		eqid 2731	. . . . . 6 ⊢ (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))) = (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))
96		fvex 6835	. . . . . 6 ⊢ (𝐺‘(𝑀 + (𝑗 − 1))) ∈ V
97	94, 95, 96	fvmpt 6929	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1))))
98	97	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐺‘(𝑀 + (𝑗 − 1))))
99		peano2nn 12134	. . . . . 6 ⊢ (𝑗 ∈ ℕ → (𝑗 + 1) ∈ ℕ)
100	99	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑗 + 1) ∈ ℕ)
101		oveq1 7353	. . . . . . . 8 ⊢ (𝑥 = (𝑗 + 1) → (𝑥 − 1) = ((𝑗 + 1) − 1))
102	101	oveq2d 7362	. . . . . . 7 ⊢ (𝑥 = (𝑗 + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + ((𝑗 + 1) − 1)))
103	102	fveq2d 6826	. . . . . 6 ⊢ (𝑥 = (𝑗 + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + ((𝑗 + 1) − 1))))
104		fvex 6835	. . . . . 6 ⊢ (𝐺‘(𝑀 + ((𝑗 + 1) − 1))) ∈ V
105	103, 95, 104	fvmpt 6929	. . . . 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 5123	. . 3 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗) < ((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘(𝑗 + 1)))
108	59	ffnd 6652	. . . . . . . 8 ⊢ (𝜑 → 𝐺 Fn 𝑍)
109		uznn0sub 12768	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → (𝑘 − 𝑀) ∈ ℕ₀)
110	11, 109	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈ ℕ₀)
111		nn0p1nn 12417	. . . . . . . . . . . 12 ⊢ ((𝑘 − 𝑀) ∈ ℕ₀ → ((𝑘 − 𝑀) + 1) ∈ ℕ)
112	110, 111	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 − 𝑀) + 1) ∈ ℕ)
113	110	nn0cnd 12441	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 − 𝑀) ∈ ℂ)
114		pncan 11363	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 − 𝑀) ∈ ℂ ∧ 1 ∈ ℂ) → (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀))
115	113, 39, 114	sylancl 586	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑘 − 𝑀) + 1) − 1) = (𝑘 − 𝑀))
116	115	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (((𝑘 − 𝑀) + 1) − 1)) = (𝑀 + (𝑘 − 𝑀)))
117		eluzelz 12739	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘𝑀) → 𝑘 ∈ ℤ)
118	117, 1	eleq2s 2849	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ)
119	118	zcnd 12575	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ)
120		pncan3 11365	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ ℂ ∧ 𝑘 ∈ ℂ) → (𝑀 + (𝑘 − 𝑀)) = 𝑘)
121	21, 119, 120	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑀 + (𝑘 − 𝑀)) = 𝑘)
122	116, 121	eqtr2d 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 = (𝑀 + (((𝑘 − 𝑀) + 1) − 1)))
123	122	fveq2d 6826	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1))))
124		oveq1 7353	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑥 − 1) = (((𝑘 − 𝑀) + 1) − 1))
125	124	oveq2d 7362	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝑀 + (𝑥 − 1)) = (𝑀 + (((𝑘 − 𝑀) + 1) − 1)))
126	125	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑥 = ((𝑘 − 𝑀) + 1) → (𝐺‘(𝑀 + (𝑥 − 1))) = (𝐺‘(𝑀 + (((𝑘 − 𝑀) + 1) − 1))))
127	126	rspceeqv 3600	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ∃𝑥 ∈ ℕ (𝐺‘𝑘) = (𝐺‘(𝑀 + (𝑥 − 1))))
129		fvex 6835	. . . . . . . . . . 11 ⊢ (𝐺‘𝑘) ∈ V
130	95	elrnmpt 5898	. . . . . . . . . . 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 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
133	132	ralrimiva 3124	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐺‘𝑘) ∈ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
134		ffnfv 7052	. . . . . . . 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 583	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝑍⟶ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
136	135	frnd 6659	. . . . . 6 ⊢ (𝜑 → ran 𝐺 ⊆ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1)))))
137	136	sscond 4096	. . . . 5 ⊢ (𝜑 → (𝑊 ∖ ran (𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))) ⊆ (𝑊 ∖ ran 𝐺))
138	137	sselda 3934	. . . 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 591	. . 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 6822	. . . . . 6 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐻‘𝑘) = (𝐻‘(𝑀 + (𝑗 − 1))))
143	69	fveq2d 6826	. . . . . 6 ⊢ (𝑘 = (𝑀 + (𝑗 − 1)) → (𝐹‘(𝐺‘𝑘)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
144	142, 143	eqeq12d 2747	. . . . 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 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
147	146	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ∀𝑘 ∈ 𝑍 (𝐻‘𝑘) = (𝐹‘(𝐺‘𝑘)))
148	144, 147, 78	rspcdva 3578	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐻‘(𝑀 + (𝑗 − 1))) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
149	93	fveq2d 6826	. . . . . 6 ⊢ (𝑥 = 𝑗 → (𝐻‘(𝑀 + (𝑥 − 1))) = (𝐻‘(𝑀 + (𝑗 − 1))))
150		fvex 6835	. . . . . 6 ⊢ (𝐻‘(𝑀 + (𝑗 − 1))) ∈ V
151	149, 33, 150	fvmpt 6929	. . . . 5 ⊢ (𝑗 ∈ ℕ → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1))))
152	151	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))‘𝑗) = (𝐻‘(𝑀 + (𝑗 − 1))))
153	98	fveq2d 6826	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐹‘((𝑥 ∈ ℕ ↦ (𝐺‘(𝑀 + (𝑥 − 1))))‘𝑗)) = (𝐹‘(𝐺‘(𝑀 + (𝑗 − 1)))))
154	148, 152, 153	3eqtr4d 2776	. . 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 15572	. 2 ⊢ (𝜑 → (seq1( + , (𝑥 ∈ ℕ ↦ (𝐻‘(𝑀 + (𝑥 − 1))))) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))
156	56, 155	bitrd 279	1 ⊢ (𝜑 → (seq𝑀( + , 𝐻) ⇝ 𝐴 ↔ seq𝑁( + , 𝐹) ⇝ 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 Vcvv 3436 ∖ cdif 3899 class class class wbr 5091 ↦ cmpt 5172 ran crn 5617 Fn wfn 6476 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ℂcc 11001 0cc0 11003 1c1 11004 + caddc 11006 < clt 11143 − cmin 11341 ℕcn 12122 ℕ₀cn0 12378 ℤcz 12465 ℤ_≥cuz 12729 ...cfz 13404 seqcseq 13905 ⇝ cli 15388

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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-oadd 8389 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-n0 12379 df-xnn0 12452 df-z 12466 df-uz 12730 df-fz 13405 df-seq 13906 df-hash 14235 df-shft 14971 df-clim 15392

This theorem is referenced by: iserodd 16744 stirlinglem5 46115

W3C validator