Theorem summodclem3 11402

Description: Lemma for summodc 11405. (Contributed by Mario Carneiro, 29-Mar-2014.) (Revised by Jim Kingdon, 9-Apr-2023.)

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
summodclem3	⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘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 summodclem3

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

Step	Hyp	Ref	Expression
1		addcl 7950	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 8108	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
4	3	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5		addass 7955	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
6	5	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ)) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
7		isummolem3.5	. . . . 5 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ))
8	7	simpld 112	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 9577	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10	8, 9	eleqtrdi 2280	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
11		isummolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
12		f1ocnv 5486	. . . . . 6 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(1...𝑀))
13	11, 12	syl 14	. . . . 5 ⊢ (𝜑 → ◡𝑓:𝐴–1-1-onto→(1...𝑀))
14		isummolem3.7	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
15		f1oco 5496	. . . . 5 ⊢ ((◡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 411	. . . 4 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17	7, 11, 14	nnf1o 11398	. . . . . . 7 ⊢ (𝜑 → 𝑁 = 𝑀)
18	17	eqcomd 2193	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
19	18	oveq2d 5904	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
20		f1oeq2 5462	. . . . 5 ⊢ ((1...𝑀) = (1...𝑁) → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
21	19, 20	syl 14	. . . 4 ⊢ (𝜑 → ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) ↔ (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀)))
22	16, 21	mpbird 167	. . 3 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
23		elnnuz 9578	. . . 4 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ≥‘1))
24		isummolem3.g	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
25		breq1 4018	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑀 ↔ 𝑚 ≤ 𝑀))
26		fveq2 5527	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
27	26	csbeq1d 3076	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
28	25, 27	ifbieq1d 3568	. . . . . . 7 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
29		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
30		elfzle2 10042	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ≤ 𝑀)
31	30	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ≤ 𝑀)
32	31	iftrued 3553	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
33		f1of 5473	. . . . . . . . . . . 12 ⊢ (𝑓:(1...𝑀)–1-1-onto→𝐴 → 𝑓:(1...𝑀)⟶𝐴)
34	11, 33	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑓:(1...𝑀)⟶𝐴)
35	34	ffvelcdmda 5664	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
36		isummo.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
37	36	ralrimiva 2560	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
38	37	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
39		nfcsb1v 3102	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
40	39	nfel1 2340	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3078	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
42	41	eleq1d 2256	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2847	. . . . . . . . . 10 ⊢ ((𝑓‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
44	35, 38, 43	sylc 62	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
45	44	adantlr 477	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
46	32, 45	eqeltrd 2264	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
47	24, 28, 29, 46	fvmptd3 5622	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
48	47, 46	eqeltrd 2264	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
49		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
50	8	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
51	50	nnzd 9388	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
52		eluzp1l 9566	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
53	51, 52	sylancom 420	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
54	49	nnzd 9388	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
55		zltnle 9313	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
56	51, 54, 55	syl2anc 411	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
57	53, 56	mpbid 147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑀)
58	57	iffalsed 3556	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = 0)
59		0cn 7963	. . . . . . . 8 ⊢ 0 ∈ ℂ
60	58, 59	eqeltrdi 2278	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
61	24, 28, 49, 60	fvmptd3 5622	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
62	61, 60	eqeltrd 2264	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) ∈ ℂ)
63		nnsplit 10151	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
64	8, 63	syl 14	. . . . . . . 8 ⊢ (𝜑 → ℕ = ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
65	64	eleq2d 2257	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1)))))
66	65	biimpa 296	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
67		elun 3288	. . . . . 6 ⊢ (𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
68	66, 67	sylib 122	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
69	48, 62, 68	mpjaodan 799	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℂ)
70	23, 69	sylan2br 288	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐺‘𝑚) ∈ ℂ)
71	17	oveq2d 5904	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = (1...𝑀))
72	71	eleq2d 2257	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
73	72	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
74	73	pm5.32i 454	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) ↔ ((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)))
75		isummolem3.4	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))
76		breq1 4018	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
77		fveq2 5527	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
78	77	csbeq1d 3076	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
79	76, 78	ifbieq1d 3568	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
80		simplr 528	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
81		elfzle2 10042	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ≤ 𝑁)
82	81	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ≤ 𝑁)
83	82	iftrued 3553	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
84		f1of 5473	. . . . . . . . . . . . 13 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
85	14, 84	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
86	85	ffvelcdmda 5664	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝐾‘𝑚) ∈ 𝐴)
87	37	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
88		nfcsb1v 3102	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
89	88	nfel1 2340	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
90		csbeq1a 3078	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
91	90	eleq1d 2256	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
92	89, 91	rspc 2847	. . . . . . . . . . 11 ⊢ ((𝐾‘𝑚) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
93	86, 87, 92	sylc 62	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
94	93	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
95	83, 94	eqeltrd 2264	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
96	75, 79, 80, 95	fvmptd3 5622	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
97	96, 95	eqeltrd 2264	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) ∈ ℂ)
98	74, 97	sylbir 135	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻‘𝑚) ∈ ℂ)
99	17	breq2d 4027	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 ≤ 𝑁 ↔ 𝑚 ≤ 𝑀))
100	99	notbid 668	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
101	100	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
102	57, 101	mpbird 167	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑁)
103	102	iffalsed 3556	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = 0)
104	103, 59	eqeltrdi 2278	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
105	75, 79, 49, 104	fvmptd3 5622	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
106	105, 104	eqeltrd 2264	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) ∈ ℂ)
107	98, 106, 68	mpjaodan 799	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℂ)
108	23, 107	sylan2br 288	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
109		f1oeq2 5462	. . . . . . . . . . 11 ⊢ ((1...𝑀) = (1...𝑁) → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
110	19, 109	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → (𝐾:(1...𝑀)–1-1-onto→𝐴 ↔ 𝐾:(1...𝑁)–1-1-onto→𝐴))
111	14, 110	mpbird 167	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
112		f1of 5473	. . . . . . . . 9 ⊢ (𝐾:(1...𝑀)–1-1-onto→𝐴 → 𝐾:(1...𝑀)⟶𝐴)
113	111, 112	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝐾:(1...𝑀)⟶𝐴)
114		fvco3 5600	. . . . . . . 8 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
115	113, 114	sylan 283	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
116	115	fveq2d 5531	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(◡𝑓‘(𝐾‘𝑖))))
117	11	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
118	113	ffvelcdmda 5664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
119		f1ocnvfv2 5792	. . . . . . 7 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
120	117, 118, 119	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
121	116, 120	eqtr2d 2221	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
122	121	csbeq1d 3076	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
123		elfznn 10068	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
124		elfzle2 10042	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
125	124	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀)
126	18	breq2d 4027	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
127	126	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
128	125, 127	mpbid 147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑁)
129	128	iftrued 3553	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
130	37	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
131		nfcsb1v 3102	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵
132	131	nfel1 2340	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ
133		csbeq1a 3078	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
134	133	eleq1d 2256	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
135	132, 134	rspc 2847	. . . . . . . 8 ⊢ ((𝐾‘𝑖) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
136	118, 130, 135	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ)
137	129, 136	eqeltrd 2264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ)
138		breq1 4018	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
139		fveq2 5527	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
140	139	csbeq1d 3076	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
141	138, 140	ifbieq1d 3568	. . . . . . 7 ⊢ (𝑛 = 𝑖 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
142	141, 75	fvmptg 5605	. . . . . 6 ⊢ ((𝑖 ∈ ℕ ∧ if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
143	123, 137, 142	syl2an2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
144	143, 129	eqtrd 2220	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
145		breq1 4018	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑛 ≤ 𝑀 ↔ ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀))
146		fveq2 5527	. . . . . . . 8 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
147	146	csbeq1d 3076	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
148	145, 147	ifbieq1d 3568	. . . . . 6 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0))
149		f1of 5473	. . . . . . . . 9 ⊢ ((◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀) → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
150	22, 149	syl 14	. . . . . . . 8 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)⟶(1...𝑀))
151	150	ffvelcdmda 5664	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
152		elfznn 10068	. . . . . . 7 ⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
153	151, 152	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ ℕ)
154		elfzle2 10042	. . . . . . . . . 10 ⊢ (((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
155	151, 154	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀)
156	155	iftrued 3553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
157	156, 122	eqtr4d 2223	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
158	157, 136	eqeltrd 2264	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ)
159	24, 148, 153, 158	fvmptd3 5622	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0))
160	159, 156	eqtrd 2220	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
161	122, 144, 160	3eqtr4d 2230	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)))
162	2, 4, 6, 10, 22, 70, 108, 161	seq3f1o 10518	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
163	18	fveq2d 5531	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
164	162, 163	eqtr3d 2222	1 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 979   = wceq 1363   ∈ wcel 2158  ∀wral 2465  ⦋csb 3069   ∪ cun 3139  ifcif 3546   class class class wbr 4015   ↦ cmpt 4076  ^◡ccnv 4637   ∘ ccom 4642  ⟶wf 5224  –1-1-onto→wf1o 5227  ‘cfv 5228  (class class class)co 5888  ℂcc 7823  0cc0 7825  1c1 7826   + caddc 7828   < clt 8006   ≤ cle 8007  ℕcn 8933  ℤcz 9267  ℤ_≥cuz 9542  ...cfz 10022  seqcseq 10459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6155  df-2nd 6156  df-recs 6320  df-frec 6406  df-1o 6431  df-er 6549  df-en 6755  df-dom 6756  df-fin 6757  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-inn 8934  df-n0 9191  df-z 9268  df-uz 9543  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-ihash 10770

This theorem is referenced by:  summodclem2a  11403  summodc  11405