Theorem summodclem3 12074

Description: Lemma for summodc 12077. (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 8257	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 277	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 8415	. . . 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 8262	. . . 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 9896	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10	8, 9	eleqtrdi 2327	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
11		isummolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
12		f1ocnv 5629	. . . . . 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 5639	. . . . 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 12070	. . . . . . 7 ⊢ (𝜑 → 𝑁 = 𝑀)
18	17	eqcomd 2240	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
19	18	oveq2d 6068	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
20		f1oeq2 5605	. . . . 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 9897	. . . 4 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ≥‘1))
24		isummolem3.g	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
25		breq1 4114	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑀 ↔ 𝑚 ≤ 𝑀))
26		fveq2 5672	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
27	26	csbeq1d 3147	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
28	25, 27	ifbieq1d 3647	. . . . . . 7 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
29		simplr 529	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
30		elfzle2 10368	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ≤ 𝑀)
31	30	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ≤ 𝑀)
32	31	iftrued 3631	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
33		f1of 5616	. . . . . . . . . . . 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 5814	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
36		isummo.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
37	36	ralrimiva 2617	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
38	37	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
39		nfcsb1v 3173	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
40	39	nfel1 2397	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3149	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
42	41	eleq1d 2303	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2917	. . . . . . . . . 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 2311	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
47	24, 28, 29, 46	fvmptd3 5773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
48	47, 46	eqeltrd 2311	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
49		simplr 529	. . . . . . 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 9705	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
52		eluzp1l 9885	. . . . . . . . . . 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 9705	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
55		zltnle 9628	. . . . . . . . . . 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 3634	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = 0)
59		0cn 8271	. . . . . . . 8 ⊢ 0 ∈ ℂ
60	58, 59	eqeltrdi 2325	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
61	24, 28, 49, 60	fvmptd3 5773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
62	61, 60	eqeltrd 2311	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) ∈ ℂ)
63		nnsplit 10478	. . . . . . . . 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 2304	. . . . . . 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 3362	. . . . . 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 806	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℂ)
70	23, 69	sylan2br 288	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐺‘𝑚) ∈ ℂ)
71	17	oveq2d 6068	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = (1...𝑀))
72	71	eleq2d 2304	. . . . . . . 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 4114	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
77		fveq2 5672	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
78	77	csbeq1d 3147	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
79	76, 78	ifbieq1d 3647	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
80		simplr 529	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
81		elfzle2 10368	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ≤ 𝑁)
82	81	adantl 277	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ≤ 𝑁)
83	82	iftrued 3631	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
84		f1of 5616	. . . . . . . . . . . . 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 5814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝐾‘𝑚) ∈ 𝐴)
87	37	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
88		nfcsb1v 3173	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
89	88	nfel1 2397	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
90		csbeq1a 3149	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
91	90	eleq1d 2303	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
92	89, 91	rspc 2917	. . . . . . . . . . 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 2311	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
96	75, 79, 80, 95	fvmptd3 5773	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
97	96, 95	eqeltrd 2311	. . . . . 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 4123	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 ≤ 𝑁 ↔ 𝑚 ≤ 𝑀))
100	99	notbid 673	. . . . . . . . . . 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 3634	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = 0)
104	103, 59	eqeltrdi 2325	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
105	75, 79, 49, 104	fvmptd3 5773	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
106	105, 104	eqeltrd 2311	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) ∈ ℂ)
107	98, 106, 68	mpjaodan 806	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℂ)
108	23, 107	sylan2br 288	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
109		f1oeq2 5605	. . . . . . . . . . 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 5616	. . . . . . . . 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 5750	. . . . . . . 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 5676	. . . . . 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 5814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
119		f1ocnvfv2 5953	. . . . . . 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 2268	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
122	121	csbeq1d 3147	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
123		elfznn 10394	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
124		elfzle2 10368	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
125	124	adantl 277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀)
126	18	breq2d 4123	. . . . . . . . . 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 3631	. . . . . . 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 3173	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵
132	131	nfel1 2397	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ
133		csbeq1a 3149	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
134	133	eleq1d 2303	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
135	132, 134	rspc 2917	. . . . . . . 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 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ)
138		breq1 4114	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
139		fveq2 5672	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
140	139	csbeq1d 3147	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
141	138, 140	ifbieq1d 3647	. . . . . . 7 ⊢ (𝑛 = 𝑖 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
142	141, 75	fvmptg 5755	. . . . . 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 598	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
144	143, 129	eqtrd 2267	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
145		breq1 4114	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑛 ≤ 𝑀 ↔ ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀))
146		fveq2 5672	. . . . . . . 8 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
147	146	csbeq1d 3147	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
148	145, 147	ifbieq1d 3647	. . . . . 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 5616	. . . . . . . . 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 5814	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
152		elfznn 10394	. . . . . . 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 10368	. . . . . . . . . 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 3631	. . . . . . . 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 2270	. . . . . . 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 2311	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ)
159	24, 148, 153, 158	fvmptd3 5773	. . . . 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 2267	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
161	122, 144, 160	3eqtr4d 2277	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)))
162	2, 4, 6, 10, 22, 70, 108, 161	seq3f1o 10886	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
163	18	fveq2d 5676	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
164	162, 163	eqtr3d 2269	1 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2205  ∀wral 2522  ⦋csb 3140   ∪ cun 3211  ifcif 3622   class class class wbr 4111   ↦ cmpt 4173  ^◡ccnv 4750   ∘ ccom 4755  ⟶wf 5350  –1-1-onto→wf1o 5353  ‘cfv 5354  (class class class)co 6052  ℂcc 8130  0cc0 8132  1c1 8133   + caddc 8135   < clt 8313   ≤ cle 8314  ℕcn 9242  ℤcz 9582  ℤ_≥cuz 9859  ...cfz 10348  seqcseq 10816

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-1o 6649  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-inn 9243  df-n0 9502  df-z 9583  df-uz 9860  df-fz 10349  df-fzo 10484  df-seqfrec 10817  df-ihash 11147

This theorem is referenced by:  summodclem2a  12075  summodc  12077