Theorem summodclem3 11181

Description: Lemma for summodc 11184. (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 7769	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) ∈ ℂ)
2	1	adantl 275	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) ∈ ℂ)
3		addcom 7923	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
4	3	adantl 275	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ)) → (𝑚 + 𝑗) = (𝑗 + 𝑚))
5		addass 7774	. . . 4 ⊢ ((𝑚 ∈ ℂ ∧ 𝑗 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑚 + 𝑗) + 𝑦) = (𝑚 + (𝑗 + 𝑦)))
6	5	adantl 275	. . 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 111	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		nnuz 9385	. . . 4 ⊢ ℕ = (ℤ≥‘1)
10	8, 9	eleqtrdi 2233	. . 3 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1))
11		isummolem3.6	. . . . . 6 ⊢ (𝜑 → 𝑓:(1...𝑀)–1-1-onto→𝐴)
12		f1ocnv 5388	. . . . . 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 5398	. . . . 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 409	. . . 4 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑁)–1-1-onto→(1...𝑀))
17	7, 11, 14	nnf1o 11177	. . . . . . 7 ⊢ (𝜑 → 𝑁 = 𝑀)
18	17	eqcomd 2146	. . . . . 6 ⊢ (𝜑 → 𝑀 = 𝑁)
19	18	oveq2d 5798	. . . . 5 ⊢ (𝜑 → (1...𝑀) = (1...𝑁))
20		f1oeq2 5365	. . . . 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 166	. . 3 ⊢ (𝜑 → (◡𝑓 ∘ 𝐾):(1...𝑀)–1-1-onto→(1...𝑀))
23		elnnuz 9386	. . . 4 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ≥‘1))
24		isummolem3.g	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
25		breq1 3940	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑀 ↔ 𝑚 ≤ 𝑀))
26		fveq2 5429	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
27	26	csbeq1d 3014	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
28	25, 27	ifbieq1d 3499	. . . . . . 7 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑀, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
29		simplr 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ∈ ℕ)
30		elfzle2 9839	. . . . . . . . . 10 ⊢ (𝑚 ∈ (1...𝑀) → 𝑚 ≤ 𝑀)
31	30	adantl 275	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → 𝑚 ≤ 𝑀)
32	31	iftrued 3486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
33		f1of 5375	. . . . . . . . . . . 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	ffvelrnda 5563	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → (𝑓‘𝑚) ∈ 𝐴)
36		isummo.2	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
37	36	ralrimiva 2508	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
38	37	adantr 274	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
39		nfcsb1v 3040	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵
40	39	nfel1 2293	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
41		csbeq1a 3016	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑓‘𝑚) → 𝐵 = ⦋(𝑓‘𝑚) / 𝑘⦌𝐵)
42	41	eleq1d 2209	. . . . . . . . . . 11 ⊢ (𝑘 = (𝑓‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
43	40, 42	rspc 2787	. . . . . . . . . 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 469	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → ⦋(𝑓‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
46	32, 45	eqeltrd 2217	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
47	24, 28, 29, 46	fvmptd3 5522	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
48	47, 46	eqeltrd 2217	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐺‘𝑚) ∈ ℂ)
49		simplr 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℕ)
50	8	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℕ)
51	50	nnzd 9196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 ∈ ℤ)
52		eluzp1l 9374	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
53	51, 52	sylancom 417	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑀 < 𝑚)
54	49	nnzd 9196	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → 𝑚 ∈ ℤ)
55		zltnle 9124	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ 𝑚 ∈ ℤ) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
56	51, 54, 55	syl2anc 409	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑀 < 𝑚 ↔ ¬ 𝑚 ≤ 𝑀))
57	53, 56	mpbid 146	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑀)
58	57	iffalsed 3489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) = 0)
59		0cn 7782	. . . . . . . 8 ⊢ 0 ∈ ℂ
60	58, 59	eqeltrdi 2231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
61	24, 28, 49, 60	fvmptd3 5522	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) = if(𝑚 ≤ 𝑀, ⦋(𝑓‘𝑚) / 𝑘⦌𝐵, 0))
62	61, 60	eqeltrd 2217	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐺‘𝑚) ∈ ℂ)
63		nnsplit 9945	. . . . . . . . 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 2210	. . . . . . 7 ⊢ (𝜑 → (𝑚 ∈ ℕ ↔ 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1)))))
66	65	biimpa 294	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))))
67		elun 3222	. . . . . 6 ⊢ (𝑚 ∈ ((1...𝑀) ∪ (ℤ≥‘(𝑀 + 1))) ↔ (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
68	66, 67	sylib 121	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑀) ∨ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))))
69	48, 62, 68	mpjaodan 788	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐺‘𝑚) ∈ ℂ)
70	23, 69	sylan2br 286	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐺‘𝑚) ∈ ℂ)
71	17	oveq2d 5798	. . . . . . . . 9 ⊢ (𝜑 → (1...𝑁) = (1...𝑀))
72	71	eleq2d 2210	. . . . . . . 8 ⊢ (𝜑 → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
73	72	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 ∈ (1...𝑁) ↔ 𝑚 ∈ (1...𝑀)))
74	73	pm5.32i 450	. . . . . 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 3940	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
77		fveq2 5429	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
78	77	csbeq1d 3014	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
79	76, 78	ifbieq1d 3499	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
80		simplr 520	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ∈ ℕ)
81		elfzle2 9839	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...𝑁) → 𝑚 ≤ 𝑁)
82	81	adantl 275	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → 𝑚 ≤ 𝑁)
83	82	iftrued 3486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
84		f1of 5375	. . . . . . . . . . . . 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	ffvelrnda 5563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → (𝐾‘𝑚) ∈ 𝐴)
87	37	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (1...𝑁)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
88		nfcsb1v 3040	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
89	88	nfel1 2293	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ
90		csbeq1a 3016	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
91	90	eleq1d 2209	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐾‘𝑚) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ))
92	89, 91	rspc 2787	. . . . . . . . . . 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 469	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
95	83, 94	eqeltrd 2217	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
96	75, 79, 80, 95	fvmptd3 5522	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
97	96, 95	eqeltrd 2217	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑁)) → (𝐻‘𝑚) ∈ ℂ)
98	74, 97	sylbir 134	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (1...𝑀)) → (𝐻‘𝑚) ∈ ℂ)
99	17	breq2d 3949	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑚 ≤ 𝑁 ↔ 𝑚 ≤ 𝑀))
100	99	notbid 657	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
101	100	ad2antrr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (¬ 𝑚 ≤ 𝑁 ↔ ¬ 𝑚 ≤ 𝑀))
102	57, 101	mpbird 166	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → ¬ 𝑚 ≤ 𝑁)
103	102	iffalsed 3489	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = 0)
104	103, 59	eqeltrdi 2231	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
105	75, 79, 49, 104	fvmptd3 5522	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
106	105, 104	eqeltrd 2217	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑚 ∈ (ℤ≥‘(𝑀 + 1))) → (𝐻‘𝑚) ∈ ℂ)
107	98, 106, 68	mpjaodan 788	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐻‘𝑚) ∈ ℂ)
108	23, 107	sylan2br 286	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
109		f1oeq2 5365	. . . . . . . . . . 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 166	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑀)–1-1-onto→𝐴)
112		f1of 5375	. . . . . . . . 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 5500	. . . . . . . 8 ⊢ ((𝐾:(1...𝑀)⟶𝐴 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
115	113, 114	sylan 281	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) = (◡𝑓‘(𝐾‘𝑖)))
116	115	fveq2d 5433	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) = (𝑓‘(◡𝑓‘(𝐾‘𝑖))))
117	11	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑓:(1...𝑀)–1-1-onto→𝐴)
118	113	ffvelrnda 5563	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) ∈ 𝐴)
119		f1ocnvfv2 5687	. . . . . . 7 ⊢ ((𝑓:(1...𝑀)–1-1-onto→𝐴 ∧ (𝐾‘𝑖) ∈ 𝐴) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
120	117, 118, 119	syl2anc 409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑓‘(◡𝑓‘(𝐾‘𝑖))) = (𝐾‘𝑖))
121	116, 120	eqtr2d 2174	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐾‘𝑖) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
122	121	csbeq1d 3014	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
123		elfznn 9865	. . . . . 6 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℕ)
124		elfzle2 9839	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ≤ 𝑀)
125	124	adantl 275	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑀)
126	18	breq2d 3949	. . . . . . . . . 10 ⊢ (𝜑 → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
127	126	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ≤ 𝑀 ↔ 𝑖 ≤ 𝑁))
128	125, 127	mpbid 146	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝑖 ≤ 𝑁)
129	128	iftrued 3486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
130	37	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑘 ∈ 𝐴 𝐵 ∈ ℂ)
131		nfcsb1v 3040	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵
132	131	nfel1 2293	. . . . . . . . 9 ⊢ Ⅎ𝑘⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ
133		csbeq1a 3016	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑖) → 𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
134	133	eleq1d 2209	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑖) → (𝐵 ∈ ℂ ↔ ⦋(𝐾‘𝑖) / 𝑘⦌𝐵 ∈ ℂ))
135	132, 134	rspc 2787	. . . . . . . 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 2217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0) ∈ ℂ)
138		breq1 3940	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → (𝑛 ≤ 𝑁 ↔ 𝑖 ≤ 𝑁))
139		fveq2 5429	. . . . . . . . 9 ⊢ (𝑛 = 𝑖 → (𝐾‘𝑛) = (𝐾‘𝑖))
140	139	csbeq1d 3014	. . . . . . . 8 ⊢ (𝑛 = 𝑖 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
141	138, 140	ifbieq1d 3499	. . . . . . 7 ⊢ (𝑛 = 𝑖 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
142	141, 75	fvmptg 5505	. . . . . 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 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = if(𝑖 ≤ 𝑁, ⦋(𝐾‘𝑖) / 𝑘⦌𝐵, 0))
144	143, 129	eqtrd 2173	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = ⦋(𝐾‘𝑖) / 𝑘⦌𝐵)
145		breq1 3940	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑛 ≤ 𝑀 ↔ ((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀))
146		fveq2 5429	. . . . . . . 8 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → (𝑓‘𝑛) = (𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)))
147	146	csbeq1d 3014	. . . . . . 7 ⊢ (𝑛 = ((◡𝑓 ∘ 𝐾)‘𝑖) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
148	145, 147	ifbieq1d 3499	. . . . . 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 5375	. . . . . . . . 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	ffvelrnda 5563	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ((◡𝑓 ∘ 𝐾)‘𝑖) ∈ (1...𝑀))
152		elfznn 9865	. . . . . . 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 9839	. . . . . . . . . 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 3486	. . . . . . . 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 2176	. . . . . . 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 2217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → if(((◡𝑓 ∘ 𝐾)‘𝑖) ≤ 𝑀, ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵, 0) ∈ ℂ)
159	24, 148, 153, 158	fvmptd3 5522	. . . . 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 2173	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)) = ⦋(𝑓‘((◡𝑓 ∘ 𝐾)‘𝑖)) / 𝑘⦌𝐵)
161	122, 144, 160	3eqtr4d 2183	. . 3 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝐻‘𝑖) = (𝐺‘((◡𝑓 ∘ 𝐾)‘𝑖)))
162	2, 4, 6, 10, 22, 70, 108, 161	seq3f1o 10308	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐺)‘𝑀))
163	18	fveq2d 5433	. 2 ⊢ (𝜑 → (seq1( + , 𝐻)‘𝑀) = (seq1( + , 𝐻)‘𝑁))
164	162, 163	eqtr3d 2175	1 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑀) = (seq1( + , 𝐻)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ⦋csb 3007   ∪ cun 3074  ifcif 3479   class class class wbr 3937   ↦ cmpt 3997  ^◡ccnv 4546   ∘ ccom 4551  ⟶wf 5127  –1-1-onto→wf1o 5130  ‘cfv 5131  (class class class)co 5782  ℂcc 7642  0cc0 7644  1c1 7645   + caddc 7647   < clt 7824   ≤ cle 7825  ℕcn 8744  ℤcz 9078  ℤ_≥cuz 9350  ...cfz 9821  seqcseq 10249

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-addcom 7744  ax-addass 7746  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-0id 7752  ax-rnegex 7753  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-frec 6296  df-1o 6321  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-inn 8745  df-n0 9002  df-z 9079  df-uz 9351  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-ihash 10554

This theorem is referenced by:  summodclem2a  11182  summodc  11184