Theorem summodclem2a 11563

Description: Lemma for summodc 11565. (Contributed by Mario Carneiro, 3-Apr-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)
isummolem2a.dc	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
isummolem2a.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
isummolem2a.h	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))
summolem2.5	⊢ (𝜑 → 𝑁 ∈ ℕ)
summolem2.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
summolem2.7	⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
summolem2.8	⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
summolem2.9	⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Assertion

Ref	Expression
summodclem2a	⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

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

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

Proof of Theorem summodclem2a

Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isummo.1	. . 3 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
2		isummo.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
3		isummolem2a.dc	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
4		summolem2.7	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ≥‘𝑀))
5		summolem2.9	. . . . . . . 8 ⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))
6		1zzd 9370	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℤ)
7		summolem2.5	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℕ)
8	7	nnzd 9464	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9	6, 8	fzfigd 10540	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
10		summolem2.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
11	9, 10	fihasheqf1od 10898	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴))
12		nnnn0 9273	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
13		hashfz1 10892	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ0 → (♯‘(1...𝑁)) = 𝑁)
14	7, 12, 13	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
15	11, 14	eqtr3d 2231	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) = 𝑁)
16	15	oveq2d 5941	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘𝐴)) = (1...𝑁))
17		isoeq4 5854	. . . . . . . . 9 ⊢ ((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
18	16, 17	syl 14	. . . . . . . 8 ⊢ (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
19	5, 18	mpbid 147	. . . . . . 7 ⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴))
20		isof1o 5857	. . . . . . 7 ⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
21	19, 20	syl 14	. . . . . 6 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
22		f1of 5507	. . . . . 6 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
23	21, 22	syl 14	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
24		nnuz 9654	. . . . . . 7 ⊢ ℕ = (ℤ≥‘1)
25	7, 24	eleqtrdi 2289	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘1))
26		eluzfz2 10124	. . . . . 6 ⊢ (𝑁 ∈ (ℤ≥‘1) → 𝑁 ∈ (1...𝑁))
27	25, 26	syl 14	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
28	23, 27	ffvelcdmd 5701	. . . 4 ⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴)
29	4, 28	sseldd 3185	. . 3 ⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ≥‘𝑀))
30	4	sselda 3184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℤ≥‘𝑀))
31		f1ocnvfv2 5828	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑛)) = 𝑛)
32	21, 31	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑛)) = 𝑛)
33		f1ocnv 5520	. . . . . . . . . . . 12 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ◡𝐾:𝐴–1-1-onto→(1...𝑁))
34		f1of 5507	. . . . . . . . . . . 12 ⊢ (◡𝐾:𝐴–1-1-onto→(1...𝑁) → ◡𝐾:𝐴⟶(1...𝑁))
35	21, 33, 34	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ◡𝐾:𝐴⟶(1...𝑁))
36	35	ffvelcdmda 5700	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (◡𝐾‘𝑛) ∈ (1...𝑁))
37		elfzle2 10120	. . . . . . . . . 10 ⊢ ((◡𝐾‘𝑛) ∈ (1...𝑁) → (◡𝐾‘𝑛) ≤ 𝑁)
38	36, 37	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (◡𝐾‘𝑛) ≤ 𝑁)
39	19	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
40		fzssuz 10157	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ (ℤ≥‘1)
41		uzssz 9638	. . . . . . . . . . . . . 14 ⊢ (ℤ≥‘1) ⊆ ℤ
42		zssre 9350	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
43	41, 42	sstri 3193	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘1) ⊆ ℝ
44	40, 43	sstri 3193	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
45		ressxr 8087	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ*
46	44, 45	sstri 3193	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℝ*
47	46	a1i 9	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (1...𝑁) ⊆ ℝ*)
48	4	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ (ℤ≥‘𝑀))
49		uzssz 9638	. . . . . . . . . . . . 13 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
50	49, 42	sstri 3193	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑀) ⊆ ℝ
51	48, 50	sstrdi 3196	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ ℝ)
52	51, 45	sstrdi 3196	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ ℝ*)
53	27	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑁 ∈ (1...𝑁))
54		leisorel 10946	. . . . . . . . . 10 ⊢ ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ* ∧ 𝐴 ⊆ ℝ*) ∧ ((◡𝐾‘𝑛) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((◡𝐾‘𝑛) ≤ 𝑁 ↔ (𝐾‘(◡𝐾‘𝑛)) ≤ (𝐾‘𝑁)))
55	39, 47, 52, 36, 53, 54	syl122anc 1258	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((◡𝐾‘𝑛) ≤ 𝑁 ↔ (𝐾‘(◡𝐾‘𝑛)) ≤ (𝐾‘𝑁)))
56	38, 55	mpbid 147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(◡𝐾‘𝑛)) ≤ (𝐾‘𝑁))
57	32, 56	eqbrtrrd 4058	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ≤ (𝐾‘𝑁))
58		eluzelz 9627	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
59	30, 58	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
60		eluzelz 9627	. . . . . . . . . 10 ⊢ ((𝐾‘𝑁) ∈ (ℤ≥‘𝑀) → (𝐾‘𝑁) ∈ ℤ)
61	29, 60	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ)
62	61	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ)
63		eluz 9631	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ≥‘𝑛) ↔ 𝑛 ≤ (𝐾‘𝑁)))
64	59, 62, 63	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ≥‘𝑛) ↔ 𝑛 ≤ (𝐾‘𝑁)))
65	57, 64	mpbird 167	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ≥‘𝑛))
66		elfzuzb 10111	. . . . . 6 ⊢ (𝑛 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑛 ∈ (ℤ≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ≥‘𝑛)))
67	30, 65, 66	sylanbrc 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (𝑀...(𝐾‘𝑁)))
68	67	ex 115	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐴 → 𝑛 ∈ (𝑀...(𝐾‘𝑁))))
69	68	ssrdv 3190	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁)))
70	1, 2, 3, 29, 69	fsum3cvg 11560	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)))
71		addlid 8182	. . . . 5 ⊢ (𝑚 ∈ ℂ → (0 + 𝑚) = 𝑚)
72	71	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (0 + 𝑚) = 𝑚)
73		addrid 8181	. . . . 5 ⊢ (𝑚 ∈ ℂ → (𝑚 + 0) = 𝑚)
74	73	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 + 0) = 𝑚)
75		addcl 8021	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 + 𝑥) ∈ ℂ)
76	75	adantl 277	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 + 𝑥) ∈ ℂ)
77		0cnd 8036	. . . 4 ⊢ (𝜑 → 0 ∈ ℂ)
78	27, 16	eleqtrrd 2276	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))
79		iftrue 3567	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) = 𝐵)
80	79	adantl 277	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) = 𝐵)
81	80, 2	eqeltrd 2273	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
82	81	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
83	82	adantlr 477	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
84		iffalse 3570	. . . . . . . . . 10 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) = 0)
85		0cn 8035	. . . . . . . . . 10 ⊢ 0 ∈ ℂ
86	84, 85	eqeltrdi 2287	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
87	86	adantl 277	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
88	3	adantlr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
89		exmiddc 837	. . . . . . . . 9 ⊢ (DECID 𝑘 ∈ 𝐴 → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
90	88, 89	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (𝑘 ∈ 𝐴 ∨ ¬ 𝑘 ∈ 𝐴))
91	83, 87, 90	mpjaodan 799	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
92		simpll 527	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝜑)
93		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ≥‘𝑀)) → ¬ 𝑘 ∈ (ℤ≥‘𝑀))
94	4	ssneld 3186	. . . . . . . . 9 ⊢ (𝜑 → (¬ 𝑘 ∈ (ℤ≥‘𝑀) → ¬ 𝑘 ∈ 𝐴))
95	92, 93, 94	sylc 62	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ≥‘𝑀)) → ¬ 𝑘 ∈ 𝐴)
96	95, 86	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ ℤ) ∧ ¬ 𝑘 ∈ (ℤ≥‘𝑀)) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
97		summolem2.6	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
98		eluzdc 9701	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → DECID 𝑘 ∈ (ℤ≥‘𝑀))
99	97, 98	sylan 283	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → DECID 𝑘 ∈ (ℤ≥‘𝑀))
100		exmiddc 837	. . . . . . . 8 ⊢ (DECID 𝑘 ∈ (ℤ≥‘𝑀) → (𝑘 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑘 ∈ (ℤ≥‘𝑀)))
101	99, 100	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ (ℤ≥‘𝑀) ∨ ¬ 𝑘 ∈ (ℤ≥‘𝑀)))
102	91, 96, 101	mpjaodan 799	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
103	102, 1	fmptd 5719	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶ℂ)
104		eluzelz 9627	. . . . 5 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) → 𝑚 ∈ ℤ)
105		ffvelcdm 5698	. . . . 5 ⊢ ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) ∈ ℂ)
106	103, 104, 105	syl2an 289	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → (𝐹‘𝑚) ∈ ℂ)
107		elnnuz 9655	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ ↔ 𝑚 ∈ (ℤ≥‘1))
108	107	biimpri 133	. . . . . . 7 ⊢ (𝑚 ∈ (ℤ≥‘1) → 𝑚 ∈ ℕ)
109	108	adantl 277	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → 𝑚 ∈ ℕ)
110		isof1o 5857	. . . . . . . . . . . 12 ⊢ (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴)
111		f1of 5507	. . . . . . . . . . . 12 ⊢ (𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
112	5, 110, 111	3syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
113	112	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝐾:(1...(♯‘𝐴))⟶𝐴)
114		1zzd 9370	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 1 ∈ ℤ)
115	15, 8	eqeltrd 2273	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ)
116	115	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (♯‘𝐴) ∈ ℤ)
117		eluzelz 9627	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℤ≥‘1) → 𝑚 ∈ ℤ)
118	117	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝑚 ∈ ℤ)
119	114, 116, 118	3jca 1179	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑚 ∈ ℤ))
120		eluzle 9630	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ℤ≥‘1) → 1 ≤ 𝑚)
121	120	ad2antlr 489	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 1 ≤ 𝑚)
122		simpr 110	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝑚 ≤ 𝑁)
123	15	breq2d 4046	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑚 ≤ (♯‘𝐴) ↔ 𝑚 ≤ 𝑁))
124	123	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝑚 ≤ (♯‘𝐴) ↔ 𝑚 ≤ 𝑁))
125	122, 124	mpbird 167	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝑚 ≤ (♯‘𝐴))
126	121, 125	jca 306	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (1 ≤ 𝑚 ∧ 𝑚 ≤ (♯‘𝐴)))
127		elfz2 10107	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (1...(♯‘𝐴)) ↔ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ ∧ 𝑚 ∈ ℤ) ∧ (1 ≤ 𝑚 ∧ 𝑚 ≤ (♯‘𝐴))))
128	119, 126, 127	sylanbrc 417	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝑚 ∈ (1...(♯‘𝐴)))
129	113, 128	ffvelcdmd 5701	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝐾‘𝑚) ∈ 𝐴)
130	129	iftrued 3569	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
131	4	ad2antrr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝐴 ⊆ (ℤ≥‘𝑀))
132	23	ad2antrr 488	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝐾:(1...𝑁)⟶𝐴)
133	16	eleq2d 2266	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑚 ∈ (1...(♯‘𝐴)) ↔ 𝑚 ∈ (1...𝑁)))
134	133	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝑚 ∈ (1...(♯‘𝐴)) ↔ 𝑚 ∈ (1...𝑁)))
135	128, 134	mpbid 147	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → 𝑚 ∈ (1...𝑁))
136	132, 135	ffvelcdmd 5701	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝐾‘𝑚) ∈ 𝐴)
137	131, 136	sseldd 3185	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝐾‘𝑚) ∈ (ℤ≥‘𝑀))
138	49, 137	sselid 3182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → (𝐾‘𝑚) ∈ ℤ)
139	102	ralrimiva 2570	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑘 ∈ ℤ if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
140	139	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → ∀𝑘 ∈ ℤ if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
141		nfv 1542	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝐾‘𝑚) ∈ 𝐴
142		nfcsb1v 3117	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝐾‘𝑚) / 𝑘⦌𝐵
143		nfcv 2339	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘0
144	141, 142, 143	nfif 3590	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0)
145	144	nfel1 2350	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ
146		eleq1 2259	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐾‘𝑚) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑚) ∈ 𝐴))
147		csbeq1a 3093	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝐾‘𝑚) → 𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
148	146, 147	ifbieq1d 3584	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑚) → if(𝑘 ∈ 𝐴, 𝐵, 0) = if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
149	148	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑚) → (if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ ↔ if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ))
150	145, 149	rspc 2862	. . . . . . . . 9 ⊢ ((𝐾‘𝑚) ∈ ℤ → (∀𝑘 ∈ ℤ if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ → if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ))
151	138, 140, 150	sylc 62	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → if((𝐾‘𝑚) ∈ 𝐴, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
152	130, 151	eqeltrrd 2274	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ 𝑚 ≤ 𝑁) → ⦋(𝐾‘𝑚) / 𝑘⦌𝐵 ∈ ℂ)
153		0cnd 8036	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) ∧ ¬ 𝑚 ≤ 𝑁) → 0 ∈ ℂ)
154	109	nnzd 9464	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → 𝑚 ∈ ℤ)
155	8	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → 𝑁 ∈ ℤ)
156		zdcle 9419	. . . . . . . 8 ⊢ ((𝑚 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑚 ≤ 𝑁)
157	154, 155, 156	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → DECID 𝑚 ≤ 𝑁)
158	152, 153, 157	ifcldadc 3591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ)
159		breq1 4037	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑛 ≤ 𝑁 ↔ 𝑚 ≤ 𝑁))
160		fveq2 5561	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝐾‘𝑛) = (𝐾‘𝑚))
161	160	csbeq1d 3091	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑚) / 𝑘⦌𝐵)
162	159, 161	ifbieq1d 3584	. . . . . . 7 ⊢ (𝑛 = 𝑚 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
163		isummolem2a.h	. . . . . . 7 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0))
164	162, 163	fvmptg 5640	. . . . . 6 ⊢ ((𝑚 ∈ ℕ ∧ if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
165	109, 158, 164	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) = if(𝑚 ≤ 𝑁, ⦋(𝐾‘𝑚) / 𝑘⦌𝐵, 0))
166	165, 158	eqeltrd 2273	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘1)) → (𝐻‘𝑚) ∈ ℂ)
167		fveqeq2 5570	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 0 ↔ (𝐹‘𝑚) = 0))
168		eldifi 3286	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))))
169		elfzelz 10117	. . . . . . . . 9 ⊢ (𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑘 ∈ ℤ)
170	168, 169	syl 14	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
171		eldifn 3287	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴)
172	171, 84	syl 14	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) = 0)
173	172, 85	eqeltrdi 2287	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
174	1	fvmpt2 5648	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
175	170, 173, 174	syl2anc 411	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
176	175, 172	eqtrd 2229	. . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = 0)
177	167, 176	vtoclga 2830	. . . . 5 ⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑚) = 0)
178	177	adantl 277	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 0)
179	112	ffvelcdmda 5700	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴)
180	179	iftrued 3569	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
181	4	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑀))
182	181, 179	sseldd 3185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ (ℤ≥‘𝑀))
183		eluzelz 9627	. . . . . . 7 ⊢ ((𝐾‘𝑥) ∈ (ℤ≥‘𝑀) → (𝐾‘𝑥) ∈ ℤ)
184	182, 183	syl 14	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ ℤ)
185		simpl 109	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝜑)
186	185, 184	jca 306	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝜑 ∧ (𝐾‘𝑥) ∈ ℤ))
187		nfv 1542	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ (𝐾‘𝑥) ∈ ℤ)
188		nfv 1542	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴
189		nfcsb1v 3117	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵
190	188, 189, 143	nfif 3590	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0)
191	190	nfel1 2350	. . . . . . . . 9 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ
192	187, 191	nfim 1586	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ (𝐾‘𝑥) ∈ ℤ) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
193		eleq1 2259	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ ℤ ↔ (𝐾‘𝑥) ∈ ℤ))
194	193	anbi2d 464	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → ((𝜑 ∧ 𝑘 ∈ ℤ) ↔ (𝜑 ∧ (𝐾‘𝑥) ∈ ℤ)))
195		eleq1 2259	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
196		csbeq1a 3093	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
197	195, 196	ifbieq1d 3584	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 0) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
198	197	eleq1d 2265	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → (if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ ↔ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ))
199	194, 198	imbi12d 234	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑥) → (((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ) ↔ ((𝜑 ∧ (𝐾‘𝑥) ∈ ℤ) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)))
200	192, 199, 102	vtoclg1f 2823	. . . . . . 7 ⊢ ((𝐾‘𝑥) ∈ 𝐴 → ((𝜑 ∧ (𝐾‘𝑥) ∈ ℤ) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ))
201	179, 186, 200	sylc 62	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
202		eleq1 2259	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → (𝑛 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
203		csbeq1 3087	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
204	202, 203	ifbieq1d 3584	. . . . . . 7 ⊢ (𝑛 = (𝐾‘𝑥) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
205		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 0)
206		nfv 1542	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
207		nfcsb1v 3117	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
208	206, 207, 143	nfif 3590	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
209		eleq1 2259	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
210		csbeq1a 3093	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
211	209, 210	ifbieq1d 3584	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
212	205, 208, 211	cbvmpt 4129	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
213	1, 212	eqtri 2217	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
214	204, 213	fvmptg 5640	. . . . . 6 ⊢ (((𝐾‘𝑥) ∈ ℤ ∧ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
215	184, 201, 214	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
216		elfznn 10146	. . . . . . . 8 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
217	216	adantl 277	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ∈ ℕ)
218		elfzle2 10120	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ≤ (♯‘𝐴))
219	218	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ (♯‘𝐴))
220	15	breq2d 4046	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ≤ (♯‘𝐴) ↔ 𝑥 ≤ 𝑁))
221	220	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝑥 ≤ (♯‘𝐴) ↔ 𝑥 ≤ 𝑁))
222	219, 221	mpbid 147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝑥 ≤ 𝑁)
223	222	iftrued 3569	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
224	180, 201	eqeltrrd 2274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)
225	223, 224	eqeltrd 2273	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
226		breq1 4037	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → (𝑛 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
227		fveq2 5561	. . . . . . . . . 10 ⊢ (𝑛 = 𝑥 → (𝐾‘𝑛) = (𝐾‘𝑥))
228	227	csbeq1d 3091	. . . . . . . . 9 ⊢ (𝑛 = 𝑥 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
229	226, 228	ifbieq1d 3584	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → if(𝑛 ≤ 𝑁, ⦋(𝐾‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
230	229, 163	fvmptg 5640	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ) → (𝐻‘𝑥) = if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
231	217, 225, 230	syl2anc 411	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = if(𝑥 ≤ 𝑁, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
232	231, 223	eqtrd 2229	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
233	180, 215, 232	3eqtr4rd 2240	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥)))
234	72, 74, 76, 77, 5, 78, 4, 106, 166, 178, 233	seq3coll 10951	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)) = (seq1( + , 𝐻)‘𝑁))
235	15, 7	eqeltrd 2273	. . . . 5 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ)
236	235, 7	jca 306	. . . 4 ⊢ (𝜑 → ((♯‘𝐴) ∈ ℕ ∧ 𝑁 ∈ ℕ))
237	16	eqcomd 2202	. . . . . 6 ⊢ (𝜑 → (1...𝑁) = (1...(♯‘𝐴)))
238		f1oeq2 5496	. . . . . 6 ⊢ ((1...𝑁) = (1...(♯‘𝐴)) → (𝑓:(1...𝑁)–1-1-onto→𝐴 ↔ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))
239	237, 238	syl 14	. . . . 5 ⊢ (𝜑 → (𝑓:(1...𝑁)–1-1-onto→𝐴 ↔ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))
240	10, 239	mpbid 147	. . . 4 ⊢ (𝜑 → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
241		isummolem2a.g	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
242	1, 2, 236, 240, 21, 241, 163	summodclem3 11562	. . 3 ⊢ (𝜑 → (seq1( + , 𝐺)‘(♯‘𝐴)) = (seq1( + , 𝐻)‘𝑁))
243	15	fveq2d 5565	. . 3 ⊢ (𝜑 → (seq1( + , 𝐺)‘(♯‘𝐴)) = (seq1( + , 𝐺)‘𝑁))
244	234, 242, 243	3eqtr2d 2235	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)) = (seq1( + , 𝐺)‘𝑁))
245	70, 244	breqtrd 4060	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ⦋csb 3084   ∖ cdif 3154   ⊆ wss 3157  ifcif 3562   class class class wbr 4034   ↦ cmpt 4095  ^◡ccnv 4663  ⟶wf 5255  –1-1-onto→wf1o 5258  ‘cfv 5259   Isom wiso 5260  (class class class)co 5925  ℂcc 7894  ℝcr 7895  0cc0 7896  1c1 7897   + caddc 7899  ℝ^*cxr 8077   < clt 8078   ≤ cle 8079  ℕcn 9007  ℕ₀cn0 9266  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  seqcseq 10556  ♯chash 10884   ⇝ cli 11460

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-1o 6483  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-n0 9267  df-z 9344  df-uz 9619  df-rp 9746  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-rsqrt 11180  df-abs 11181  df-clim 11461

This theorem is referenced by:  summodclem2  11564  zsumdc  11566