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Theorem prodmolem2a 15282

Description: Lemma for prodmo 15284. (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 𝐵,𝑗 𝑓,𝑗,𝑘 𝑗,𝐺 𝑗,𝑘,𝜑 𝑗,𝐾,𝑘 𝑗,𝑀,𝑘 𝑗,𝑁,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 𝑚 𝑥 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 13335	. . . . . . . . . . 11 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
6		prodmolem2.8	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑓:(1...𝑁)–1-1-onto→𝐴)
7	5, 6	hasheqf1od 13708	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...𝑁)) = (♯‘𝐴))
8		prodmolem2.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ)
9	8	nnnn0d 11949	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
10		hashfz1 13700	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
12	7, 11	eqtr3d 2858	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐴) = 𝑁)
13	12	oveq2d 7166	. . . . . . . 8 ⊢ (𝜑 → (1...(♯‘𝐴)) = (1...𝑁))
14		isoeq4 7067	. . . . . . . 8 ⊢ ((1...(♯‘𝐴)) = (1...𝑁) → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) ↔ 𝐾 Isom < , < ((1...𝑁), 𝐴)))
16	4, 15	mpbid 234	. . . . . 6 ⊢ (𝜑 → 𝐾 Isom < , < ((1...𝑁), 𝐴))
17		isof1o 7070	. . . . . 6 ⊢ (𝐾 Isom < , < ((1...𝑁), 𝐴) → 𝐾:(1...𝑁)–1-1-onto→𝐴)
18		f1of 6610	. . . . . 6 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → 𝐾:(1...𝑁)⟶𝐴)
19	16, 17, 18	3syl 18	. . . . 5 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶𝐴)
20		nnuz 12275	. . . . . . 7 ⊢ ℕ = (ℤ_≥‘1)
21	8, 20	eleqtrdi 2923	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘1))
22		eluzfz2 12909	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘1) → 𝑁 ∈ (1...𝑁))
23	21, 22	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
24	19, 23	ffvelrnd 6847	. . . 4 ⊢ (𝜑 → (𝐾‘𝑁) ∈ 𝐴)
25	3, 24	sseldd 3968	. . 3 ⊢ (𝜑 → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑀))
26	3	sselda 3967	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (ℤ_≥‘𝑀))
27	16, 17	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐾:(1...𝑁)–1-1-onto→𝐴)
28		f1ocnvfv2 7028	. . . . . . . . 9 ⊢ ((𝐾:(1...𝑁)–1-1-onto→𝐴 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
29	27, 28	sylan 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) = 𝑗)
30		f1ocnv 6622	. . . . . . . . . . . 12 ⊢ (𝐾:(1...𝑁)–1-1-onto→𝐴 → ^◡𝐾:𝐴–1-1-onto→(1...𝑁))
31		f1of 6610	. . . . . . . . . . . 12 ⊢ (^◡𝐾:𝐴–1-1-onto→(1...𝑁) → ^◡𝐾:𝐴⟶(1...𝑁))
32	27, 30, 31	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ^◡𝐾:𝐴⟶(1...𝑁))
33	32	ffvelrnda 6846	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ∈ (1...𝑁))
34		elfzle2 12905	. . . . . . . . . 10 ⊢ ((^◡𝐾‘𝑗) ∈ (1...𝑁) → (^◡𝐾‘𝑗) ≤ 𝑁)
35	33, 34	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (^◡𝐾‘𝑗) ≤ 𝑁)
36	16	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐾 Isom < , < ((1...𝑁), 𝐴))
37		fzssuz 12942	. . . . . . . . . . . . 13 ⊢ (1...𝑁) ⊆ (ℤ_≥‘1)
38		uzssz 12258	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘1) ⊆ ℤ
39		zssre 11982	. . . . . . . . . . . . . 14 ⊢ ℤ ⊆ ℝ
40	38, 39	sstri 3976	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘1) ⊆ ℝ
41	37, 40	sstri 3976	. . . . . . . . . . . 12 ⊢ (1...𝑁) ⊆ ℝ
42		ressxr 10679	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℝ^*
43	41, 42	sstri 3976	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ ℝ^*
44	43	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (1...𝑁) ⊆ ℝ^*)
45		uzssz 12258	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
46	45, 39	sstri 3976	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ
47	46, 42	sstri 3976	. . . . . . . . . . . 12 ⊢ (ℤ_≥‘𝑀) ⊆ ℝ^*
48	3, 47	sstrdi 3979	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ ℝ^*)
49	48	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝐴 ⊆ ℝ^*)
50	23	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑁 ∈ (1...𝑁))
51		leisorel 13812	. . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((^◡𝐾‘𝑗) ≤ 𝑁 ↔ (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁)))
53	35, 52	mpbid 234	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘(^◡𝐾‘𝑗)) ≤ (𝐾‘𝑁))
54	29, 53	eqbrtrrd 5083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ≤ (𝐾‘𝑁))
55	3, 45	sstrdi 3979	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
56	55	sselda 3967	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ ℤ)
57		eluzelz 12247	. . . . . . . . . 10 ⊢ ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑀) → (𝐾‘𝑁) ∈ ℤ)
58	25, 57	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝐾‘𝑁) ∈ ℤ)
59	58	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ ℤ)
60		eluz 12251	. . . . . . . 8 ⊢ ((𝑗 ∈ ℤ ∧ (𝐾‘𝑁) ∈ ℤ) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
61	56, 59, 60	syl2anc 586	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → ((𝐾‘𝑁) ∈ (ℤ_≥‘𝑗) ↔ 𝑗 ≤ (𝐾‘𝑁)))
62	54, 61	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗))
63		elfzuzb 12896	. . . . . 6 ⊢ (𝑗 ∈ (𝑀...(𝐾‘𝑁)) ↔ (𝑗 ∈ (ℤ_≥‘𝑀) ∧ (𝐾‘𝑁) ∈ (ℤ_≥‘𝑗)))
64	26, 62, 63	sylanbrc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐴) → 𝑗 ∈ (𝑀...(𝐾‘𝑁)))
65	64	ex 415	. . . 4 ⊢ (𝜑 → (𝑗 ∈ 𝐴 → 𝑗 ∈ (𝑀...(𝐾‘𝑁))))
66	65	ssrdv 3973	. . 3 ⊢ (𝜑 → 𝐴 ⊆ (𝑀...(𝐾‘𝑁)))
67	1, 2, 25, 66	fprodcvg 15278	. 2 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)))
68		mulid2 10634	. . . . 5 ⊢ (𝑚 ∈ ℂ → (1 · 𝑚) = 𝑚)
69	68	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (1 · 𝑚) = 𝑚)
70		mulid1 10633	. . . . 5 ⊢ (𝑚 ∈ ℂ → (𝑚 · 1) = 𝑚)
71	70	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℂ) → (𝑚 · 1) = 𝑚)
72		mulcl 10615	. . . . 5 ⊢ ((𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑚 · 𝑥) ∈ ℂ)
73	72	adantl 484	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑚 · 𝑥) ∈ ℂ)
74		1cnd 10630	. . . 4 ⊢ (𝜑 → 1 ∈ ℂ)
75	23, 13	eleqtrrd 2916	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (1...(♯‘𝐴)))
76		iftrue 4473	. . . . . . . . . . 11 ⊢ (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
77	76	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 𝐵)
78	77, 2	eqeltrd 2913	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
79	78	ex 415	. . . . . . . 8 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ))
80		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
81		ax-1cn 10589	. . . . . . . . 9 ⊢ 1 ∈ ℂ
82	80, 81	eqeltrdi 2921	. . . . . . . 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 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
85	84, 1	fmptd 6873	. . . . 5 ⊢ (𝜑 → 𝐹:ℤ⟶ℂ)
86		elfzelz 12902	. . . . 5 ⊢ (𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴))) → 𝑚 ∈ ℤ)
87		ffvelrn 6844	. . . . 5 ⊢ ((𝐹:ℤ⟶ℂ ∧ 𝑚 ∈ ℤ) → (𝐹‘𝑚) ∈ ℂ)
88	85, 86, 87	syl2an 597	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...(𝐾‘(♯‘𝐴)))) → (𝐹‘𝑚) ∈ ℂ)
89		fveqeq2 6674	. . . . . 6 ⊢ (𝑘 = 𝑚 → ((𝐹‘𝑘) = 1 ↔ (𝐹‘𝑚) = 1))
90		fzssz 12903	. . . . . . . . 9 ⊢ (𝑀...(𝐾‘(♯‘𝐴))) ⊆ ℤ
91		eldifi 4103	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ (𝑀...(𝐾‘(♯‘𝐴))))
92	90, 91	sseldi 3965	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → 𝑘 ∈ ℤ)
93		eldifn 4104	. . . . . . . . . 10 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → ¬ 𝑘 ∈ 𝐴)
94	93, 80	syl 17	. . . . . . . . 9 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) = 1)
95	94, 81	eqeltrdi 2921	. . . . . . . 8 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ)
96	1	fvmpt2 6774	. . . . . . . 8 ⊢ ((𝑘 ∈ ℤ ∧ if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
97	92, 95, 96	syl2anc 586	. . . . . . 7 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = if(𝑘 ∈ 𝐴, 𝐵, 1))
98	97, 94	eqtrd 2856	. . . . . 6 ⊢ (𝑘 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑘) = 1)
99	89, 98	vtoclga 3574	. . . . 5 ⊢ (𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴) → (𝐹‘𝑚) = 1)
100	99	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ((𝑀...(𝐾‘(♯‘𝐴))) ∖ 𝐴)) → (𝐹‘𝑚) = 1)
101		isof1o 7070	. . . . . . . 8 ⊢ (𝐾 Isom < , < ((1...(♯‘𝐴)), 𝐴) → 𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴)
102		f1of 6610	. . . . . . . 8 ⊢ (𝐾:(1...(♯‘𝐴))–1-1-onto→𝐴 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
103	4, 101, 102	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐾:(1...(♯‘𝐴))⟶𝐴)
104	103	ffvelrnda 6846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ 𝐴)
105	104	iftrued 4475	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
106	55	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → 𝐴 ⊆ ℤ)
107	106, 104	sseldd 3968	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐾‘𝑥) ∈ ℤ)
108		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑘𝜑
109		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑘(𝐾‘𝑥) ∈ 𝐴
110		nfcsb1v 3907	. . . . . . . . . . 11 ⊢ Ⅎ𝑘⦋(𝐾‘𝑥) / 𝑘⦌𝐵
111		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑘1
112	109, 110, 111	nfif 4496	. . . . . . . . . 10 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1)
113	112	nfel1 2994	. . . . . . . . 9 ⊢ Ⅎ𝑘if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ
114	108, 113	nfim 1893	. . . . . . . 8 ⊢ Ⅎ𝑘(𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
115		fvex 6678	. . . . . . . 8 ⊢ (𝐾‘𝑥) ∈ V
116		eleq1 2900	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → (𝑘 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
117		csbeq1a 3897	. . . . . . . . . . 11 ⊢ (𝑘 = (𝐾‘𝑥) → 𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
118	116, 117	ifbieq1d 4490	. . . . . . . . . 10 ⊢ (𝑘 = (𝐾‘𝑥) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
119	118	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑘 = (𝐾‘𝑥) → (if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ ↔ if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ))
120	119	imbi2d 343	. . . . . . . 8 ⊢ (𝑘 = (𝐾‘𝑥) → ((𝜑 → if(𝑘 ∈ 𝐴, 𝐵, 1) ∈ ℂ) ↔ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)))
121	114, 115, 120, 83	vtoclf 3559	. . . . . . 7 ⊢ (𝜑 → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
122	121	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1) ∈ ℂ)
123		eleq1 2900	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → (𝑛 ∈ 𝐴 ↔ (𝐾‘𝑥) ∈ 𝐴))
124		csbeq1 3886	. . . . . . . 8 ⊢ (𝑛 = (𝐾‘𝑥) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
125	123, 124	ifbieq1d 4490	. . . . . . 7 ⊢ (𝑛 = (𝐾‘𝑥) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
126		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑛if(𝑘 ∈ 𝐴, 𝐵, 1)
127		nfv 1911	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ 𝐴
128		nfcsb1v 3907	. . . . . . . . . 10 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
129	127, 128, 111	nfif 4496	. . . . . . . . 9 ⊢ Ⅎ𝑘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1)
130		eleq1 2900	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → (𝑘 ∈ 𝐴 ↔ 𝑛 ∈ 𝐴))
131		csbeq1a 3897	. . . . . . . . . 10 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
132	130, 131	ifbieq1d 4490	. . . . . . . . 9 ⊢ (𝑘 = 𝑛 → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
133	126, 129, 132	cbvmpt 5160	. . . . . . . 8 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
134	1, 133	eqtri 2844	. . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 1))
135	125, 134	fvmptg 6761	. . . . . 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))
136	107, 122, 135	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐹‘(𝐾‘𝑥)) = if((𝐾‘𝑥) ∈ 𝐴, ⦋(𝐾‘𝑥) / 𝑘⦌𝐵, 1))
137		elfznn 12930	. . . . . 6 ⊢ (𝑥 ∈ (1...(♯‘𝐴)) → 𝑥 ∈ ℕ)
138	105, 122	eqeltrrd 2914	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ)
139		fveq2 6665	. . . . . . . 8 ⊢ (𝑗 = 𝑥 → (𝐾‘𝑗) = (𝐾‘𝑥))
140	139	csbeq1d 3887	. . . . . . 7 ⊢ (𝑗 = 𝑥 → ⦋(𝐾‘𝑗) / 𝑘⦌𝐵 = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
141		prodmolem2.4	. . . . . . 7 ⊢ 𝐻 = (𝑗 ∈ ℕ ↦ ⦋(𝐾‘𝑗) / 𝑘⦌𝐵)
142	140, 141	fvmptg 6761	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ ⦋(𝐾‘𝑥) / 𝑘⦌𝐵 ∈ ℂ) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
143	137, 138, 142	syl2an2 684	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = ⦋(𝐾‘𝑥) / 𝑘⦌𝐵)
144	105, 136, 143	3eqtr4rd 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(♯‘𝐴))) → (𝐻‘𝑥) = (𝐹‘(𝐾‘𝑥)))
145	69, 71, 73, 74, 4, 75, 3, 88, 100, 144	seqcoll 13816	. . 3 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐻)‘𝑁))
146		prodmo.3	. . . 4 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
147	8, 8	jca 514	. . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ ∧ 𝑁 ∈ ℕ))
148	1, 2, 146, 141, 147, 6, 27	prodmolem3 15281	. . 3 ⊢ (𝜑 → (seq1( · , 𝐺)‘𝑁) = (seq1( · , 𝐻)‘𝑁))
149	145, 148	eqtr4d 2859	. 2 ⊢ (𝜑 → (seq𝑀( · , 𝐹)‘(𝐾‘𝑁)) = (seq1( · , 𝐺)‘𝑁))
150	67, 149	breqtrd 5085	1 ⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ (seq1( · , 𝐺)‘𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⦋csb 3883 ∖ cdif 3933 ⊆ wss 3936 ifcif 4467 class class class wbr 5059 ↦ cmpt 5139 ^◡ccnv 5549 ⟶wf 6346 –1-1-onto→wf1o 6349 ‘cfv 6350 Isom wiso 6351 (class class class)co 7150 Fincfn 8503 ℂcc 10529 ℝcr 10530 1c1 10532 · cmul 10536 ℝ^*cxr 10668 < clt 10669 ≤ cle 10670 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ...cfz 12886 seqcseq 13363 ♯chash 13684 ⇝ cli 14835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839

This theorem is referenced by: prodmolem2 15283 zprod 15285

W3C validator