prodfzo03 - Mathbox for Thierry Arnoux

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Theorem prodfzo03 31867

Description: A product of three factors, indexed starting with zero. (Contributed by Thierry Arnoux, 14-Dec-2021.)

**Hypotheses**
Ref	Expression
prodfzo03.1	⊢ (𝑘 = 0 → 𝐷 = 𝐴)
prodfzo03.2	⊢ (𝑘 = 1 → 𝐷 = 𝐵)
prodfzo03.3	⊢ (𝑘 = 2 → 𝐷 = 𝐶)
prodfzo03.a	⊢ ((𝜑 ∧ 𝑘 ∈ (0..^3)) → 𝐷 ∈ ℂ)

**Assertion**
Ref	Expression
prodfzo03	⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝐶,𝑘 𝜑,𝑘 Allowed substitution hint: 𝐷(𝑘)

**Proof of Theorem prodfzo03**
Step	Hyp	Ref	Expression
1		fzodisjsn 13067	. . . . 5 ⊢ ((0..^2) ∩ {2}) = ∅
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ((0..^2) ∩ {2}) = ∅)
3		2p1e3 11771	. . . . . . 7 ⊢ (2 + 1) = 3
4	3	oveq2i 7159	. . . . . 6 ⊢ (0..^(2 + 1)) = (0..^3)
5		2eluzge0 12285	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
6		fzosplitsn 13137	. . . . . . 7 ⊢ (2 ∈ (ℤ_≥‘0) → (0..^(2 + 1)) = ((0..^2) ∪ {2}))
7	5, 6	ax-mp 5	. . . . . 6 ⊢ (0..^(2 + 1)) = ((0..^2) ∪ {2})
8	4, 7	eqtr3i 2844	. . . . 5 ⊢ (0..^3) = ((0..^2) ∪ {2})
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) = ((0..^2) ∪ {2}))
10		fzofi 13334	. . . . 5 ⊢ (0..^3) ∈ Fin
11	10	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) ∈ Fin)
12		prodfzo03.a	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^3)) → 𝐷 ∈ ℂ)
13	2, 9, 11, 12	fprodsplit 15312	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷))
14		0ne1 11700	. . . . . 6 ⊢ 0 ≠ 1
15		disjsn2 4640	. . . . . 6 ⊢ (0 ≠ 1 → ({0} ∩ {1}) = ∅)
16	14, 15	mp1i 13	. . . . 5 ⊢ (𝜑 → ({0} ∩ {1}) = ∅)
17		fzo0to2pr 13114	. . . . . . 7 ⊢ (0..^2) = {0, 1}
18		df-pr 4562	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
19	17, 18	eqtri 2842	. . . . . 6 ⊢ (0..^2) = ({0} ∪ {1})
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) = ({0} ∪ {1}))
21		fzofi 13334	. . . . . 6 ⊢ (0..^2) ∈ Fin
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) ∈ Fin)
23		2z 12006	. . . . . . . . 9 ⊢ 2 ∈ ℤ
24		3z 12007	. . . . . . . . 9 ⊢ 3 ∈ ℤ
25		2re 11703	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
26		3re 11709	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
27		2lt3 11801	. . . . . . . . . 10 ⊢ 2 < 3
28	25, 26, 27	ltleii 10755	. . . . . . . . 9 ⊢ 2 ≤ 3
29		eluz2 12241	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3))
30	23, 24, 28, 29	mpbir3an 1336	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
31		fzoss2 13057	. . . . . . . 8 ⊢ (3 ∈ (ℤ_≥‘2) → (0..^2) ⊆ (0..^3))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (0..^2) ⊆ (0..^3)
33	32	sseli 3961	. . . . . 6 ⊢ (𝑘 ∈ (0..^2) → 𝑘 ∈ (0..^3))
34	33, 12	sylan2 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^2)) → 𝐷 ∈ ℂ)
35	16, 20, 22, 34	fprodsplit 15312	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (0..^2)𝐷 = (∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷))
36	35	oveq1d 7163	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷) = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
37	13, 36	eqtrd 2854	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
38		snfi 8586	. . . . 5 ⊢ {0} ∈ Fin
39	38	a1i 11	. . . 4 ⊢ (𝜑 → {0} ∈ Fin)
40		velsn 4575	. . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
41		prodfzo03.1	. . . . . . 7 ⊢ (𝑘 = 0 → 𝐷 = 𝐴)
42	41	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 = 𝐴)
43		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 = 𝐴)
44	12	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 ∈ ℂ)
45	43, 44	eqeltrrd 2912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐴 ∈ ℂ)
46		c0ex 10627	. . . . . . . . . . . 12 ⊢ 0 ∈ V
47	46	tpid1 4696	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1, 2}
48		fzo0to3tp 13115	. . . . . . . . . . 11 ⊢ (0..^3) = {0, 1, 2}
49	47, 48	eleqtrri 2910	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
50		eqid 2819	. . . . . . . . . 10 ⊢ 𝐴 = 𝐴
51	41	eqeq1d 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝐷 = 𝐴 ↔ 𝐴 = 𝐴))
52	51	rspcev 3621	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^3) ∧ 𝐴 = 𝐴) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
53	49, 50, 52	mp2an 690	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐴
54	53	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
55	45, 54	r19.29a 3287	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
56	55	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 ∈ ℂ)
57	42, 56	eqeltrd 2911	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 ∈ ℂ)
58	40, 57	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝐷 ∈ ℂ)
59	39, 58	fprodcl 15298	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 ∈ ℂ)
60		snfi 8586	. . . . 5 ⊢ {1} ∈ Fin
61	60	a1i 11	. . . 4 ⊢ (𝜑 → {1} ∈ Fin)
62		velsn 4575	. . . . 5 ⊢ (𝑘 ∈ {1} ↔ 𝑘 = 1)
63		prodfzo03.2	. . . . . . 7 ⊢ (𝑘 = 1 → 𝐷 = 𝐵)
64	63	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 = 𝐵)
65		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 = 𝐵)
66	12	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 ∈ ℂ)
67	65, 66	eqeltrrd 2912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐵 ∈ ℂ)
68		1ex 10629	. . . . . . . . . . . 12 ⊢ 1 ∈ V
69	68	tpid2 4698	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1, 2}
70	69, 48	eleqtrri 2910	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
71		eqid 2819	. . . . . . . . . 10 ⊢ 𝐵 = 𝐵
72	63	eqeq1d 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐷 = 𝐵 ↔ 𝐵 = 𝐵))
73	72	rspcev 3621	. . . . . . . . . 10 ⊢ ((1 ∈ (0..^3) ∧ 𝐵 = 𝐵) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
74	70, 71, 73	mp2an 690	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐵
75	74	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
76	67, 75	r19.29a 3287	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
77	76	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐵 ∈ ℂ)
78	64, 77	eqeltrd 2911	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
79	62, 78	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {1}) → 𝐷 ∈ ℂ)
80	61, 79	fprodcl 15298	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 ∈ ℂ)
81		snfi 8586	. . . . 5 ⊢ {2} ∈ Fin
82	81	a1i 11	. . . 4 ⊢ (𝜑 → {2} ∈ Fin)
83		velsn 4575	. . . . 5 ⊢ (𝑘 ∈ {2} ↔ 𝑘 = 2)
84		prodfzo03.3	. . . . . . 7 ⊢ (𝑘 = 2 → 𝐷 = 𝐶)
85	84	adantl 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 = 𝐶)
86		simpr 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 = 𝐶)
87	12	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 ∈ ℂ)
88	86, 87	eqeltrrd 2912	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐶 ∈ ℂ)
89		2ex 11706	. . . . . . . . . . . 12 ⊢ 2 ∈ V
90	89	tpid3 4701	. . . . . . . . . . 11 ⊢ 2 ∈ {0, 1, 2}
91	90, 48	eleqtrri 2910	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
92		eqid 2819	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
93	84	eqeq1d 2821	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐷 = 𝐶 ↔ 𝐶 = 𝐶))
94	93	rspcev 3621	. . . . . . . . . 10 ⊢ ((2 ∈ (0..^3) ∧ 𝐶 = 𝐶) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
95	91, 92, 94	mp2an 690	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐶
96	95	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
97	88, 96	r19.29a 3287	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
98	97	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐶 ∈ ℂ)
99	85, 98	eqeltrd 2911	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 ∈ ℂ)
100	83, 99	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {2}) → 𝐷 ∈ ℂ)
101	82, 100	fprodcl 15298	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 ∈ ℂ)
102	59, 80, 101	mulassd 10656	. 2 ⊢ (𝜑 → ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷) = (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)))
103		0nn0 11904	. . . . 5 ⊢ 0 ∈ ℕ₀
104	103	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
105	41	prodsn 15308	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ 𝐴 ∈ ℂ) → ∏𝑘 ∈ {0}𝐷 = 𝐴)
106	104, 55, 105	syl2anc 586	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 = 𝐴)
107		1nn0 11905	. . . . . 6 ⊢ 1 ∈ ℕ₀
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
109	63	prodsn 15308	. . . . 5 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {1}𝐷 = 𝐵)
110	108, 76, 109	syl2anc 586	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 = 𝐵)
111		2nn0 11906	. . . . . 6 ⊢ 2 ∈ ℕ₀
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
113	84	prodsn 15308	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {2}𝐷 = 𝐶)
114	112, 97, 113	syl2anc 586	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 = 𝐶)
115	110, 114	oveq12d 7166	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷) = (𝐵 · 𝐶))
116	106, 115	oveq12d 7166	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)) = (𝐴 · (𝐵 · 𝐶)))
117	37, 102, 116	3eqtrd 2858	1 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ≠ wne 3014 ∃wrex 3137 ∪ cun 3932 ∩ cin 3933 ⊆ wss 3934 ∅c0 4289 {csn 4559 {cpr 4561 {ctp 4563 class class class wbr 5057 ‘cfv 6348 (class class class)co 7148 Fincfn 8501 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 ≤ cle 10668 2c2 11684 3c3 11685 ℕ₀cn0 11889 ℤcz 11973 ℤ_≥cuz 12235 ..^cfzo 13025 ∏cprod 15251

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-inf2 9096 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-n0 11890 df-z 11974 df-uz 12236 df-rp 12382 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252

This theorem is referenced by: circlevma 31906 circlemethhgt 31907

W3C validator