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Theorem prodmolem2a 16107

Description: Lemma for prodmo 16109. (Contributed by Scott Fenton, 4-Dec-2017.)

**Hypotheses**
Ref	Expression
prodmo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
prodmo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
prodmo.3	⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
prodmolem2.4	⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
prodmolem2.5	⊢ (𝜑 → 𝑁 ∈ ℕ)
prodmolem2.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
prodmolem2.7	⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
prodmolem2.8	⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
prodmolem2.9	⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))

**Assertion**
Ref	Expression
prodmolem2a	⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

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

Proof of Theorem prodmolem2a Dummy variables 𝑛 𝑥 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodmo.1	. . 3 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
2		prodmo.2	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
3		prodmolem2.7	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (ℤ_≥‘𝑀))
4		prodmolem2.9	. . . . . . 7 ⊢ (𝜑 → 𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴))
5		fzfid 14120	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		prodmolem2.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
7	5, 6	hasheqf1od 14501	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴))
8		prodmolem2.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 12671	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		hashfz1 14494	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
11	9, 10	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
12	7, 11	eqtr3d 2797	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) = 𝑁)
13	12	oveq2d 7437	. . . . . . . 8 ⊢ (𝜑 → (1...(♯‘𝐴)) = (1...𝑁))
14		isoeq4 7329	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
15	13, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
16	4, 15	mpbid 235	. . . . . 6 ⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴))
17		isof1o 7332	. . . . . 6 ⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
18		f1of 6825	. . . . . 6 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
19	16, 17, 18	3syl 19	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
20		nnuz 13008	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
21	8, 20	eleqtrdi 2870	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
22		eluzfz2 13669	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
23	21, 22	syl 18	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
24	19, 23	ffvelcdmd 7086	. . . 4 ⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴)
25	3, 24	sseldd 3909	. . 3 ⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑀))
26	3	sselda 3908	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (ℤ_≥‘𝑀))
27	16, 17	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
28		f1ocnvfv2 7286	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
29	27, 28	sylan 592	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
30		f1ocnv 6838	. . . . . . . . . . . 12 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ^◡𝐾:𝐴–1-1-onto→(1...𝑁))
31		f1of 6825	. . . . . . . . . . . 12 ⊢ (^◡𝐾:𝐴–1-1-onto→(1...𝑁) → ^◡𝐾:𝐴⟶(1...𝑁))
32	27, 30, 31	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐾:𝐴⟶(1...𝑁))
33	32	ffvelcdmda 7085	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ∈ (1...𝑁))
34		elfzle2 13665	. . . . . . . . . 10 ⊢ ((^◡𝐾‘𝑗) ∈ (1...𝑁) → (^◡𝐾‘𝑗) ≤ 𝑁)
35	33, 34	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ≤ 𝑁)
36	16	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
37		fzssuz 13703	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ (ℤ_≥‘1)
38		uzssz 12990	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
39		zssre 12704	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
40	38, 39	sstri 3917	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘1) ⊆ ℝ
41	37, 40	sstri 3917	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
42		ressxr 11357	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
43	41, 42	sstri 3917	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℝ^*
44	43	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ⊆ ℝ^*)
45		uzssz 12990	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
46	45, 39	sstri 3917	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
47	46, 42	sstri 3917	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ^*
48	3, 47	sstrdi 3920	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
49	48	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
50	23	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑁 ∈ (1...𝑁))
51		leisorel 14609	. . . . . . . . . 10 ⊢ ((𝐾 Isom < , < ((1...𝑁), 𝐴) ∧ ((1...𝑁) ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^*) ∧ ((^◡𝐾‘𝑗) ∈ (1...𝑁) ∧ 𝑁 ∈ (1...𝑁))) → ((^◡𝐾‘𝑗) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁)))
52	36, 44, 49, 33, 50, 51	syl122anc 1406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((^◡𝐾‘𝑗) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁)))
53	35, 52	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁))
54	29, 53	eqbrtrrd 5128	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ≤ (𝐾‘𝑁))
55	3, 45	sstrdi 3920	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
56	55	sselda 3908	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℤ)
57		eluzelz 12979	. . . . . . . . . 10 ⊢ ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑀) → (𝐾‘𝑁) ∈ ℤ)
58	25, 57	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ)
59	58	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ)
60		eluz 12983	. . . . . . . 8 ⊢ ((𝑗 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
61	56, 59, 60	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
62	54, 61	mpbird 260	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗))
63		elfzuzb 13654	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑗 ∈ (ℤ_≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗)))
64	26, 62, 63	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (𝑀...(𝐾‘𝑁)))
65	64	ex 418	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ (𝑀...(𝐾‘𝑁))))
66	65	ssrdv 3914	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁)))
67	1, 2, 25, 66	fprodcvg 16103	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)))
68		mullid 11311	. . . . 5 ⊢ (𝑚 ∈ ℂ → (1 · 𝑚) = 𝑚)
69	68	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (1 · 𝑚) = 𝑚)
70		mulrid 11310	. . . . 5 ⊢ (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚)
71	70	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚)
72		mulcl 11288	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 · 𝑥) ∈ ℂ)
73	72	adantl 487	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 · 𝑥) ∈ ℂ)
74		1cnd 11306	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
75	23, 13	eleqtrrd 2863	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))
76		iftrue 4487	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
77	76	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
78	77, 2	eqeltrd 2860	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
79	78	ex 418	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ))
80		iffalse 4490	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
81		ax-1cn 11262	. . . . . . . . 9 ⊢ 1 ∈ ℂ
82	80, 81	eqeltrdi 2868	. . . . . . . 8 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
83	79, 82	pm2.61d1 182	. . . . . . 7 ⊢ (𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
84	83	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
85	84, 1	fmptd 7115	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶ℂ)
86		elfzelz 13660	. . . . 5 ⊢ (𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑚 ∈ ℤ)
87		ffvelcdm 7082	. . . . 5 ⊢ ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) ∈ ℂ)
88	85, 86, 87	syl2an 608	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴)))) → (𝐹‘𝑚) ∈ ℂ)
89		fveqeq2 6895	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 1 ↔ (𝐹‘𝑚) = 1))
90		eldifi 4067	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))))
91	90	elfzelzd 13661	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
92		eldifn 4068	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴)
93	92, 80	syl 18	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
94	93, 81	eqeltrdi 2868	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
95	1	fvmpt2 7006	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
96	91, 94, 95	syl2anc 596	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
97	96, 93	eqtrd 2795	. . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = 1)
98	89, 97	vtoclga 3536	. . . . 5 ⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑚) = 1)
99	98	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 1)
100		isof1o 7332	. . . . . . . 8 ⊢ (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴)
101		f1of 6825	. . . . . . . 8 ⊢ (𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
102	4, 100, 101	3syl 19	. . . . . . 7 ⊢ (𝜑 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
103	102	ffvelcdmda 7085	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴)
104	103	iftrued 4489	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
105	55	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ ℤ)
106	105, 103	sseldd 3909	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ ℤ)
107		nfv 1946	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
108		nfv 1946	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴
109		nfcsb1v 3870	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵
110		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑘1
111	108, 109, 110	nfif 4512	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)
112	111	nfel1 2938	. . . . . . . . 9 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ
113	107, 112	nfim 1928	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
114		fvex 6899	. . . . . . . 8 ⊢ (𝐾‘𝑥) ∈ V
115		eleq1 2848	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
116		csbeq1a 3860	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
117	115, 116	ifbieq1d 4506	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
118	117	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → (if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ ↔ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ))
119	118	imbi2d 343	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑥) → ((𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) ↔ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)))
120	113, 114, 119, 83	vtoclf 3525	. . . . . . 7 ⊢ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
121	120	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
122		eleq1 2848	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → (𝑛 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
123		csbeq1 3849	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
124	122, 123	ifbieq1d 4506	. . . . . . 7 ⊢ (𝑛 = (𝐾‘𝑥) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
125		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 1)
126		nfv 1946	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
127		nfcsb1v 3870	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
128	126, 127, 110	nfif 4512	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1)
129		eleq1 2848	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
130		csbeq1a 3860	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
131	129, 130	ifbieq1d 4506	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
132	125, 128, 131	cbvmpt 5206	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
133	1, 132	eqtri 2783	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
134	124, 133	fvmptg 6992	. . . . . 6 ⊢ (((𝐾‘𝑥) ∈ ℤ ∧ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
135	106, 121, 134	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
136		elfznn 13691	. . . . . 6 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
137	104, 121	eqeltrrd 2861	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)
138		fveq2 6886	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝐾‘𝑗) = (𝐾‘𝑥))
139	138	csbeq1d 3850	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
140		prodmolem2.4	. . . . . . 7 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
141	139, 140	fvmptg 6992	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
142	136, 137, 141	syl2an2 699	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
143	104, 135, 142	3eqtr4rd 2806	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥)))
144	69, 71, 73, 74, 4, 75, 3, 88, 99, 143	seqcoll 14613	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐻)‘𝑁))
145		prodmo.3	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
146	8, 8	jca 521	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ))
147	1, 2, 145, 140, 146, 6, 27	prodmolem3 16106	. . 3 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑁) = (seq1( · , 𝐻)‘𝑁))
148	144, 147	eqtr4d 2798	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐺)‘𝑁))
149	67, 148	breqtrd 5130	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ⦋csb 3846 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 class class class wbr 5102 ↦ cmpt 5185 ^◡ccnv 5649 ⟶wf 6535 –1-1-onto→wf1o 6538 ‘cfv 6539 Isom wiso 6540 (class class class)co 7421 Fincfn 8977 ℂcc 11202 ℝcr 11203 1c1 11205 · cmul 11209 ℝ^*cxr 11346 < clt 11347 ≤ cle 11348 ℕcn 12339 ℕ₀cn0 12610 ℤcz 12697 ℤ_≥cuz 12969 ...cfz 13643 seqcseq 14148 ♯chash 14478 ⇝ cli 15657

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-inf2 9646 ax-cnex 11260 ax-resscn 11261 ax-1cn 11262 ax-icn 11263 ax-addcl 11264 ax-addrcl 11265 ax-mulcl 11266 ax-mulrcl 11267 ax-mulcom 11268 ax-addass 11269 ax-mulass 11270 ax-distr 11271 ax-i2m1 11272 ax-1ne0 11273 ax-1rid 11274 ax-rnegex 11275 ax-rrecex 11276 ax-cnre 11277 ax-pre-lttri 11278 ax-pre-lttrn 11279 ax-pre-ltadd 11280 ax-pre-mulgt0 11281

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-isom 6548 df-riota 7378 df-ov 7424 df-oprab 7425 df-mpo 7426 df-om 7879 df-1st 8002 df-2nd 8003 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-1o 8480 df-er 8721 df-en 8978 df-dom 8979 df-sdom 8980 df-fin 8981 df-card 10024 df-pnf 11349 df-mnf 11350 df-xr 11351 df-ltxr 11352 df-le 11353 df-sub 11547 df-neg 11548 df-div 11978 df-nn 12340 df-2 12409 df-n0 12611 df-z 12698 df-uz 12970 df-rp 13125 df-fz 13644 df-fzo 13793 df-seq 14149 df-exp 14209 df-hash 14479 df-cj 15272 df-re 15273 df-im 15274 df-sqrt 15408 df-abs 15409 df-clim 15661

This theorem is used by: prodmolem2 16108 zprod 16110

W3C validator