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Theorem summolem2a 15619

Description: Lemma for summo 15621. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Mario Carneiro, 20-Apr-2014.)

**Hypotheses**
Ref	Expression
summo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
summo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summo.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
summolem2.4	⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
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
summolem2a	⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

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

Proof of Theorem summolem2a Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		summo.1	. . 3 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
2		summo.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
3		summolem2.7	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
4		summolem2.9	. . . . . . . 8 ⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5		fzfid 13877	. . . . . . . . . . . 12 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		summolem2.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
7	5, 6	hasheqf1od 14257	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴))
8		summolem2.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9		nnnn0 12385	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ₀)
10		hashfz1 14250	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
11	8, 9, 10	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
12	7, 11	eqtr3d 2768	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐴) = 𝑁)
13	12	oveq2d 7362	. . . . . . . . 9 ⊢ (𝜑 → (1...(♯‘𝐴)) = (1...𝑁))
14		isoeq4 7254	. . . . . . . . 9 ⊢ ((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
16	4, 15	mpbid 232	. . . . . . 7 ⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴))
17		isof1o 7257	. . . . . . 7 ⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
18	16, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
19		f1of 6763	. . . . . 6 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
20	18, 19	syl 17	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
21		nnuz 12772	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
22	8, 21	eleqtrdi 2841	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
23		eluzfz2 13429	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
24	22, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
25	20, 24	ffvelcdmd 7018	. . . 4 ⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴)
26	3, 25	sseldd 3935	. . 3 ⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑀))
27	3	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (ℤ_≥‘𝑀))
28		f1ocnvfv2 7211	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑛)) = 𝑛)
29	18, 28	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑛)) = 𝑛)
30		f1ocnv 6775	. . . . . . . . . . . 12 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ^◡𝐾:𝐴–1-1-onto→(1...𝑁))
31		f1of 6763	. . . . . . . . . . . 12 ⊢ (^◡𝐾:𝐴–1-1-onto→(1...𝑁) → ^◡𝐾:𝐴⟶(1...𝑁))
32	18, 30, 31	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐾:𝐴⟶(1...𝑁))
33	32	ffvelcdmda 7017	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (^◡𝐾‘𝑛) ∈ (1...𝑁))
34		elfzle2 13425	. . . . . . . . . 10 ⊢ ((^◡𝐾‘𝑛) ∈ (1...𝑁) → (^◡𝐾‘𝑛) ≤ 𝑁)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (^◡𝐾‘𝑛) ≤ 𝑁)
36	16	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
37		fzssuz 13462	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ (ℤ_≥‘1)
38		uzssz 12750	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
39		zssre 12472	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
40	38, 39	sstri 3944	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘1) ⊆ ℝ
41	37, 40	sstri 3944	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
42		ressxr 11153	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
43	41, 42	sstri 3944	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℝ^*
44	43	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (1...𝑁) ⊆ ℝ^*)
45	3	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ (ℤ_≥‘𝑀))
46		uzssz 12750	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
47	46, 39	sstri 3944	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
48	45, 47	sstrdi 3947	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ ℝ)
49	48, 42	sstrdi 3947	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
50	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑁 ∈ (1...𝑁))
51		leisorel 14364	. . . . . . . . . 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)))
52	36, 44, 49, 33, 50, 51	syl122anc 1381	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((^◡𝐾‘𝑛) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑛)) ≤ (𝐾‘𝑁)))
53	35, 52	mpbid 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑛)) ≤ (𝐾‘𝑁))
54	29, 53	eqbrtrrd 5115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ≤ (𝐾‘𝑁))
55		eluzelz 12739	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → 𝑛 ∈ ℤ)
56	27, 55	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ ℤ)
57		eluzelz 12739	. . . . . . . . . 10 ⊢ ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑀) → (𝐾‘𝑁) ∈ ℤ)
58	26, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ)
59	58	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ)
60		eluz 12743	. . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑛) ↔ 𝑛 ≤ (𝐾‘𝑁)))
61	56, 59, 60	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑛) ↔ 𝑛 ≤ (𝐾‘𝑁)))
62	54, 61	mpbird 257	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑛))
63		elfzuzb 13415	. . . . . 6 ⊢ (𝑛 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑛 ∈ (ℤ_≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ_≥‘𝑛)))
64	27, 62, 63	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ (𝑀...(𝐾‘𝑁)))
65	64	ex 412	. . . 4 ⊢ (𝜑 → (𝑛 ∈ 𝐴 → 𝑛 ∈ (𝑀...(𝐾‘𝑁))))
66	65	ssrdv 3940	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁)))
67	1, 2, 26, 66	fsumcvg 15616	. 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)))
68		addlid 11293	. . . . 5 ⊢ (𝑚 ∈ ℂ → (0 + 𝑚) = 𝑚)
69	68	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (0 + 𝑚) = 𝑚)
70		addrid 11290	. . . . 5 ⊢ (𝑚 ∈ ℂ → (𝑚 + 0) = 𝑚)
71	70	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 + 0) = 𝑚)
72		addcl 11085	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 + 𝑥) ∈ ℂ)
73	72	adantl 481	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 + 𝑥) ∈ ℂ)
74		0cnd 11102	. . . 4 ⊢ (𝜑 → 0 ∈ ℂ)
75	24, 13	eleqtrrd 2834	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))
76		iftrue 4481	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) = 𝐵)
77	76	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) = 𝐵)
78	77, 2	eqeltrd 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
79	78	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ))
80		iffalse 4484	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) = 0)
81		0cn 11101	. . . . . . . . 9 ⊢ 0 ∈ ℂ
82	80, 81	eqeltrdi 2839	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
83	79, 82	pm2.61d1 180	. . . . . . 7 ⊢ (𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
84	83	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
85	84, 1	fmptd 7047	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶ℂ)
86		elfzelz 13421	. . . . 5 ⊢ (𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑚 ∈ ℤ)
87		ffvelcdm 7014	. . . . 5 ⊢ ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) ∈ ℂ)
88	85, 86, 87	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴)))) → (𝐹‘𝑚) ∈ ℂ)
89		fveqeq2 6831	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 0 ↔ (𝐹‘𝑚) = 0))
90		eldifi 4081	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))))
91	90	elfzelzd 13422	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
92		eldifn 4082	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴)
93	92, 80	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) = 0)
94	93, 81	eqeltrdi 2839	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ)
95	1	fvmpt2 6940	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
96	91, 94, 95	syl2anc 584	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 0))
97	96, 93	eqtrd 2766	. . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = 0)
98	89, 97	vtoclga 3532	. . . . 5 ⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑚) = 0)
99	98	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 0)
100		isof1o 7257	. . . . . . . 8 ⊢ (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴)
101		f1of 6763	. . . . . . . 8 ⊢ (𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
102	4, 100, 101	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
103	102	ffvelcdmda 7017	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴)
104	103	iftrued 4483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
105	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ (ℤ_≥‘𝑀))
106	105, 103	sseldd 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ (ℤ_≥‘𝑀))
107		eluzelz 12739	. . . . . . 7 ⊢ ((𝐾‘𝑥) ∈ (ℤ_≥‘𝑀) → (𝐾‘𝑥) ∈ ℤ)
108	106, 107	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ ℤ)
109		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
110		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴
111		nfcsb1v 3874	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵
112		nfcv 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑘0
113	110, 111, 112	nfif 4506	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0)
114	113	nfel1 2911	. . . . . . . . 9 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ
115	109, 114	nfim 1897	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
116		fvex 6835	. . . . . . . 8 ⊢ (𝐾‘𝑥) ∈ V
117		eleq1 2819	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
118		csbeq1a 3864	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
119	117, 118	ifbieq1d 4500	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 0) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
120	119	eleq1d 2816	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → (if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ ↔ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ))
121	120	imbi2d 340	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑥) → ((𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 0) ∈ ℂ) ↔ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)))
122	115, 116, 121, 83	vtoclf 3519	. . . . . . 7 ⊢ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
123	122	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0) ∈ ℂ)
124		eleq1 2819	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → (𝑛 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
125		csbeq1 3853	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
126	124, 125	ifbieq1d 4500	. . . . . . 7 ⊢ (𝑛 = (𝐾‘𝑥) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
127		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 0)
128		nfv 1915	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
129		nfcsb1v 3874	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
130	128, 129, 112	nfif 4506	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
131		eleq1 2819	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
132		csbeq1a 3864	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
133	131, 132	ifbieq1d 4500	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
134	127, 130, 133	cbvmpt 5193	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
135	1, 134	eqtri 2754	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
136	126, 135	fvmptg 6927	. . . . . 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))
137	108, 123, 136	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 0))
138		elfznn 13450	. . . . . 6 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
139	104, 123	eqeltrrd 2832	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)
140		fveq2 6822	. . . . . . . 8 ⊢ (𝑛 = 𝑥 → (𝐾‘𝑛) = (𝐾‘𝑥))
141	140	csbeq1d 3854	. . . . . . 7 ⊢ (𝑛 = 𝑥 → ⦋(𝐾‘𝑛) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
142		summolem2.4	. . . . . . 7 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ⦋(𝐾‘𝑛) / 𝑘⦌𝐵)
143	141, 142	fvmptg 6927	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
144	138, 139, 143	syl2an2 686	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
145	104, 137, 144	3eqtr4rd 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥)))
146	69, 71, 73, 74, 4, 75, 3, 88, 99, 145	seqcoll 14368	. . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)) = (seq1( + , 𝐻)‘𝑁))
147		summo.3	. . . 4 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
148	8, 8	jca 511	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ))
149	1, 2, 147, 142, 148, 6, 18	summolem3 15618	. . 3 ⊢ (𝜑 → (seq1( + , 𝐺)‘𝑁) = (seq1( + , 𝐻)‘𝑁))
150	146, 149	eqtr4d 2769	. 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝐾‘𝑁)) = (seq1( + , 𝐺)‘𝑁))
151	67, 150	breqtrd 5117	1 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ⦋csb 3850 ∖ cdif 3899 ⊆ wss 3902 ifcif 4475 class class class wbr 5091 ↦ cmpt 5172 ^◡ccnv 5615 ⟶wf 6477 –1-1-onto→wf1o 6480 ‘cfv 6481 Isom wiso 6482 (class class class)co 7346 Fincfn 8869 ℂcc 11001 ℝcr 11002 0cc0 11003 1c1 11004 + caddc 11006 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144 ℕcn 12122 ℕ₀cn0 12378 ℤcz 12465 ℤ_≥cuz 12729 ...cfz 13404 seqcseq 13905 ♯chash 14234 ⇝ cli 15388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-inf2 9531 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-card 9829 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-n0 12379 df-z 12466 df-uz 12730 df-rp 12888 df-fz 13405 df-fzo 13552 df-seq 13906 df-exp 13966 df-hash 14235 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-clim 15392

This theorem is referenced by: summolem2 15620 zsum 15622

W3C validator