Theorem isummolem3 10770

Description: Lemma for isummo 10773. (Contributed by Mario Carneiro, 29-Mar-2014.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
isummolem3.5	⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
isummolem3.6	⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
isummolem3.7	⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
isummolem3.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
isummolem3.4	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
isummolem3	⊢ (𝜑 → (seq1( + , 𝐺, ℂ)‘𝑀) = (seq1( + , 𝐻, ℂ)‘𝑁))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝑘,𝑁,𝑛   𝜑,𝑘,𝑛   𝑘,𝑀,𝑛   𝐵,𝑛   𝑘,𝐾,𝑛   𝑓,𝑘,𝑛

Allowed substitution hints:   𝜑(𝑓)   𝐴(𝑓)   𝐵(𝑓,𝑘)   𝐹(𝑓,𝑘)   𝐺(𝑓,𝑘,𝑛)   𝐻(𝑓,𝑘,𝑛)   𝐾(𝑓)   𝑀(𝑓)   𝑁(𝑓)

Proof of Theorem isummolem3

Dummy variables 𝑖 𝑗 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 7467	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 271	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 7619	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
4	3	adantl 271	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5		addass 7472	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
6	5	adantl 271	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7		isummolem3.5	. . . . . 6 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 110	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 9054	. . . . 5 ⊢ ℕ = (ℤ≥‘1)
10	8, 9	syl6eleq 2180	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
11		isummolem3.6	. . . . . . 7 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
12		f1ocnv 5266	. . . . . . 7 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(1...𝑀))
13	11, 12	syl 14	. . . . . 6 ⊢ (𝜑 → ◡𝑓:𝐴–1-1-onto→(1...𝑀))
14		isummolem3.7	. . . . . 6 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
15		f1oco 5276	. . . . . 6 ⊢ ((◡𝑓:𝐴–1-1-onto→(1...𝑀) ∧ 𝐾:(1...𝑁)–1-1-onto→𝐴) → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
16	13, 14, 15	syl2anc 403	. . . . 5 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17		isummo.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
18		isummo.2	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
19	17, 18, 7, 11, 14	isummolemnm 10769	. . . . . . . 8 ⊢ (𝜑 → 𝑁 = 𝑀)
20	19	eqcomd 2093	. . . . . . 7 ⊢ (𝜑 → 𝑀 = 𝑁)
21	20	oveq2d 5668	. . . . . 6 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
22		f1oeq2 5245	. . . . . 6 ⊢ ((1...𝑀) = (1...𝑁) → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
23	21, 22	syl 14	. . . . 5 ⊢ (𝜑 → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
24	16, 23	mpbird 165	. . . 4 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
25		elnnuz 9055	. . . . 5 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ≥‘1))
26		simplr 497	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
27		elfzle2 9442	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ≤ 𝑀)
28	27	adantl 271	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ≤ 𝑀)
29	28	iftrued 3400	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
30		f1of 5253	. . . . . . . . . . . . 13 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
31	11, 30	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
32	31	ffvelrnda 5434	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
33	18	ralrimiva 2446	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
34	33	adantr 270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
35		nfcsb1v 2963	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
36	35	nfel1 2239	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
37		csbeq1a 2941	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
38	37	eleq1d 2156	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
39	36, 38	rspc 2716	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
40	32, 34, 39	sylc 61	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
41	40	adantlr 461	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
42	29, 41	eqeltrd 2164	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
43		breq1 3848	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑀 ↔ 𝑚 ≤ 𝑀))
44		fveq2 5305	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
45	44	csbeq1d 2939	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
46	43, 45	ifbieq1d 3413	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
47		isummolem3.g	. . . . . . . . 9 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
48	46, 47	fvmptg 5380	. . . . . . . 8 ⊢ ((𝑚 ∈ ℕ ∧ if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
49	26, 42, 48	syl2anc 403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
50	49, 42	eqeltrd 2164	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
51		simplr 497	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
52	8	ad2antrr 472	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
53	52	nnzd 8867	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
54		eluzp1l 9043	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
55	53, 54	sylancom 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
56	51	nnzd 8867	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
57		zltnle 8796	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
58	53, 56, 57	syl2anc 403	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
59	55, 58	mpbid 145	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑀)
60	59	iffalsed 3403	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = 0)
61		0cn 7480	. . . . . . . . 9 ⊢ 0 ∈ ℂ
62	60, 61	syl6eqel 2178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
63	51, 62, 48	syl2anc 403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
64	63, 62	eqeltrd 2164	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) ∈ ℂ)
65		nnsplit 9548	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
66	8, 65	syl 14	. . . . . . . . 9 ⊢ (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
67	66	eleq2d 2157	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1)))))
68	67	biimpa 290	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
69		elun 3141	. . . . . . 7 ⊢ (𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
70	68, 69	sylib 120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
71	50, 64, 70	mpjaodan 747	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℂ)
72	25, 71	sylan2br 282	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐺‘𝑚) ∈ ℂ)
73	19	oveq2d 5668	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) = (1...𝑀))
74	73	eleq2d 2157	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
75	74	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
76	75	pm5.32i 442	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)))
77		simplr 497	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
78		elfzle2 9442	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ≤ 𝑁)
79	78	adantl 271	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ≤ 𝑁)
80	79	iftrued 3400	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
81		f1of 5253	. . . . . . . . . . . . . 14 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
82	14, 81	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
83	82	ffvelrnda 5434	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝐾‘𝑚) ∈ 𝐴)
84	33	adantr 270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
85		nfcsb1v 2963	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
86	85	nfel1 2239	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
87		csbeq1a 2941	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
88	87	eleq1d 2156	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
89	86, 88	rspc 2716	. . . . . . . . . . . 12 ⊢ ((𝐾‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
90	83, 84, 89	sylc 61	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
91	90	adantlr 461	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
92	80, 91	eqeltrd 2164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
93		breq1 3848	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
94		fveq2 5305	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
95	94	csbeq1d 2939	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
96	93, 95	ifbieq1d 3413	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
97		isummolem3.4	. . . . . . . . . 10 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))
98	96, 97	fvmptg 5380	. . . . . . . . 9 ⊢ ((𝑚 ∈ ℕ ∧ if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
99	77, 92, 98	syl2anc 403	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
100	99, 92	eqeltrd 2164	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) ∈ ℂ)
101	76, 100	sylbir 133	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻‘𝑚) ∈ ℂ)
102	19	breq2d 3857	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑚 ≤ 𝑁 ↔ 𝑚 ≤ 𝑀))
103	102	notbid 627	. . . . . . . . . . . 12 ⊢ (𝜑 → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
104	103	ad2antrr 472	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
105	59, 104	mpbird 165	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑁)
106	105	iffalsed 3403	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = 0)
107	106, 61	syl6eqel 2178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
108	51, 107, 98	syl2anc 403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
109	108, 107	eqeltrd 2164	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) ∈ ℂ)
110	101, 109, 70	mpjaodan 747	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℂ)
111	25, 110	sylan2br 282	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
112		f1oeq2 5245	. . . . . . . . . . . 12 ⊢ ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
113	21, 112	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
114	14, 113	mpbird 165	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
115		f1of 5253	. . . . . . . . . 10 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
116	114, 115	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
117		fvco3 5375	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
118	116, 117	sylan 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
119	118	fveq2d 5309	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(◡𝑓‘(𝐾‘𝑖))))
120	11	adantr 270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
121	116	ffvelrnda 5434	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
122		f1ocnvfv2 5557	. . . . . . . 8 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
123	120, 121, 122	syl2anc 403	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
124	119, 123	eqtr2d 2121	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
125	124	csbeq1d 2939	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
126		elfznn 9468	. . . . . . 7 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
127		elfzle2 9442	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
128	127	adantl 271	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀)
129	20	breq2d 3857	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
130	129	adantr 270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
131	128, 130	mpbid 145	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑁)
132	131	iftrued 3400	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
133	33	adantr 270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
134		nfcsb1v 2963	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵
135	134	nfel1 2239	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ
136		csbeq1a 2941	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
137	136	eleq1d 2156	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
138	135, 137	rspc 2716	. . . . . . . . 9 ⊢ ((𝐾‘𝑖) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
139	121, 133, 138	sylc 61	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ)
140	132, 139	eqeltrd 2164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ)
141		breq1 3848	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
142		fveq2 5305	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
143	142	csbeq1d 2939	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
144	141, 143	ifbieq1d 3413	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
145	144, 97	fvmptg 5380	. . . . . . 7 ⊢ ((𝑖 ∈ ℕ ∧ if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
146	126, 140, 145	syl2an2 561	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
147	146, 132	eqtrd 2120	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
148		f1of 5253	. . . . . . . . . 10 ⊢ ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
149	24, 148	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
150	149	ffvelrnda 5434	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
151		elfznn 9468	. . . . . . . 8 ⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
152	150, 151	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
153		elfzle2 9442	. . . . . . . . . . 11 ⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
154	150, 153	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
155	154	iftrued 3400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
156	155, 125	eqtr4d 2123	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
157	156, 139	eqeltrd 2164	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ)
158		breq1 3848	. . . . . . . . 9 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑛 ≤ 𝑀 ↔ ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀))
159		fveq2 5305	. . . . . . . . . 10 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
160	159	csbeq1d 2939	. . . . . . . . 9 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
161	158, 160	ifbieq1d 3413	. . . . . . . 8 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0))
162	161, 47	fvmptg 5380	. . . . . . 7 ⊢ ((((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ ∧ if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0))
163	152, 157, 162	syl2anc 403	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0))
164	163, 155	eqtrd 2120	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
165	125, 147, 164	3eqtr4d 2130	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)))
166	2, 4, 6, 10, 24, 72, 111, 165	seq3f1o 9933	. . 3 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
167		1zzd 8777	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
168	167, 111	iseqseq3 9902	. . . 4 ⊢ (𝜑 → seq1( + , 𝐻, ℂ) = seq1( + , 𝐻))
169	168	fveq1d 5307	. . 3 ⊢ (𝜑 → (seq1( + , 𝐻, ℂ)‘𝑀) = (seq1( + , 𝐻)‘𝑀))
170	167, 72	iseqseq3 9902	. . . 4 ⊢ (𝜑 → seq1( + , 𝐺, ℂ) = seq1( + , 𝐺))
171	170	fveq1d 5307	. . 3 ⊢ (𝜑 → (seq1( + , 𝐺, ℂ)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
172	166, 169, 171	3eqtr4d 2130	. 2 ⊢ (𝜑 → (seq1( + , 𝐻, ℂ)‘𝑀) = (seq1( + , 𝐺, ℂ)‘𝑀))
173	20	fveq2d 5309	. 2 ⊢ (𝜑 → (seq1( + , 𝐻, ℂ)‘𝑀) = (seq1( + , 𝐻, ℂ)‘𝑁))
174	172, 173	eqtr3d 2122	1 ⊢ (𝜑 → (seq1( + , 𝐺, ℂ)‘𝑀) = (seq1( + , 𝐻, ℂ)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   ∨ wo 664   ∧ w3a 924   = wceq 1289   ∈ wcel 1438  ∀wral 2359  ⦋csb 2933   ∪ cun 2997  ifcif 3393   class class class wbr 3845   ↦ cmpt 3899  ^◡ccnv 4437   ∘ ccom 4442  ⟶wf 5011  –1-1-onto→wf1o 5014  ‘cfv 5015  (class class class)co 5652  ℂcc 7348  0cc0 7350  1c1 7351   + caddc 7353   < clt 7522   ≤ cle 7523  ℕcn 8422  ℤcz 8750  ℤ_≥cuz 9019  ...cfz 9424  seqcseq4 9851  seqcseq 9852

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-1re 7439  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-distr 7449  ax-i2m1 7450  ax-0lt1 7451  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456  ax-pre-ltirr 7457  ax-pre-ltwlin 7458  ax-pre-lttrn 7459  ax-pre-apti 7460  ax-pre-ltadd 7461

This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-1o 6181  df-er 6292  df-en 6458  df-dom 6459  df-fin 6460  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-sub 7655  df-neg 7656  df-inn 8423  df-n0 8674  df-z 8751  df-uz 9020  df-fz 9425  df-fzo 9554  df-iseq 9853  df-seq3 9854  df-ihash 10184

This theorem is referenced by:  isummolem2a  10771  isummo  10773