prodfzo03 - Mathbox for Thierry Arnoux

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Theorem prodfzo03 33615

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 13670	. . . . 5 ⊢ ((0..^2) ∩ {2}) = ∅
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ((0..^2) ∩ {2}) = ∅)
3		2p1e3 12354	. . . . . . 7 ⊢ (2 + 1) = 3
4	3	oveq2i 7420	. . . . . 6 ⊢ (0..^(2 + 1)) = (0..^3)
5		2eluzge0 12877	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
6		fzosplitsn 13740	. . . . . . 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 2763	. . . . 5 ⊢ (0..^3) = ((0..^2) ∪ {2})
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) = ((0..^2) ∪ {2}))
10		fzofi 13939	. . . . 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 15910	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷))
14		0ne1 12283	. . . . . 6 ⊢ 0 ≠ 1
15		disjsn2 4717	. . . . . 6 ⊢ (0 ≠ 1 → ({0} ∩ {1}) = ∅)
16	14, 15	mp1i 13	. . . . 5 ⊢ (𝜑 → ({0} ∩ {1}) = ∅)
17		fzo0to2pr 13717	. . . . . . 7 ⊢ (0..^2) = {0, 1}
18		df-pr 4632	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
19	17, 18	eqtri 2761	. . . . . 6 ⊢ (0..^2) = ({0} ∪ {1})
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) = ({0} ∪ {1}))
21		fzofi 13939	. . . . . 6 ⊢ (0..^2) ∈ Fin
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) ∈ Fin)
23		2z 12594	. . . . . . . . 9 ⊢ 2 ∈ ℤ
24		3z 12595	. . . . . . . . 9 ⊢ 3 ∈ ℤ
25		2re 12286	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
26		3re 12292	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
27		2lt3 12384	. . . . . . . . . 10 ⊢ 2 < 3
28	25, 26, 27	ltleii 11337	. . . . . . . . 9 ⊢ 2 ≤ 3
29		eluz2 12828	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3))
30	23, 24, 28, 29	mpbir3an 1342	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
31		fzoss2 13660	. . . . . . . 8 ⊢ (3 ∈ (ℤ_≥‘2) → (0..^2) ⊆ (0..^3))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (0..^2) ⊆ (0..^3)
33	32	sseli 3979	. . . . . 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 15910	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (0..^2)𝐷 = (∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷))
36	35	oveq1d 7424	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷) = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
37	13, 36	eqtrd 2773	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
38		snfi 9044	. . . . 5 ⊢ {0} ∈ Fin
39	38	a1i 11	. . . 4 ⊢ (𝜑 → {0} ∈ Fin)
40		velsn 4645	. . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
41		prodfzo03.1	. . . . . . 7 ⊢ (𝑘 = 0 → 𝐷 = 𝐴)
42	41	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 = 𝐴)
43		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 = 𝐴)
44	12	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 ∈ ℂ)
45	43, 44	eqeltrrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐴 ∈ ℂ)
46		c0ex 11208	. . . . . . . . . . . 12 ⊢ 0 ∈ V
47	46	tpid1 4773	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1, 2}
48		fzo0to3tp 13718	. . . . . . . . . . 11 ⊢ (0..^3) = {0, 1, 2}
49	47, 48	eleqtrri 2833	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
50		eqid 2733	. . . . . . . . . 10 ⊢ 𝐴 = 𝐴
51	41	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝐷 = 𝐴 ↔ 𝐴 = 𝐴))
52	51	rspcev 3613	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^3) ∧ 𝐴 = 𝐴) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
53	49, 50, 52	mp2an 691	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐴
54	53	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
55	45, 54	r19.29a 3163	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
56	55	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 ∈ ℂ)
57	42, 56	eqeltrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 ∈ ℂ)
58	40, 57	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝐷 ∈ ℂ)
59	39, 58	fprodcl 15896	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 ∈ ℂ)
60		snfi 9044	. . . . 5 ⊢ {1} ∈ Fin
61	60	a1i 11	. . . 4 ⊢ (𝜑 → {1} ∈ Fin)
62		velsn 4645	. . . . 5 ⊢ (𝑘 ∈ {1} ↔ 𝑘 = 1)
63		prodfzo03.2	. . . . . . 7 ⊢ (𝑘 = 1 → 𝐷 = 𝐵)
64	63	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 = 𝐵)
65		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 = 𝐵)
66	12	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 ∈ ℂ)
67	65, 66	eqeltrrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐵 ∈ ℂ)
68		1ex 11210	. . . . . . . . . . . 12 ⊢ 1 ∈ V
69	68	tpid2 4775	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1, 2}
70	69, 48	eleqtrri 2833	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
71		eqid 2733	. . . . . . . . . 10 ⊢ 𝐵 = 𝐵
72	63	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐷 = 𝐵 ↔ 𝐵 = 𝐵))
73	72	rspcev 3613	. . . . . . . . . 10 ⊢ ((1 ∈ (0..^3) ∧ 𝐵 = 𝐵) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
74	70, 71, 73	mp2an 691	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐵
75	74	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
76	67, 75	r19.29a 3163	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
77	76	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐵 ∈ ℂ)
78	64, 77	eqeltrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
79	62, 78	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {1}) → 𝐷 ∈ ℂ)
80	61, 79	fprodcl 15896	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 ∈ ℂ)
81		snfi 9044	. . . . 5 ⊢ {2} ∈ Fin
82	81	a1i 11	. . . 4 ⊢ (𝜑 → {2} ∈ Fin)
83		velsn 4645	. . . . 5 ⊢ (𝑘 ∈ {2} ↔ 𝑘 = 2)
84		prodfzo03.3	. . . . . . 7 ⊢ (𝑘 = 2 → 𝐷 = 𝐶)
85	84	adantl 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 = 𝐶)
86		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 = 𝐶)
87	12	adantr 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 ∈ ℂ)
88	86, 87	eqeltrrd 2835	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐶 ∈ ℂ)
89		2ex 12289	. . . . . . . . . . . 12 ⊢ 2 ∈ V
90	89	tpid3 4778	. . . . . . . . . . 11 ⊢ 2 ∈ {0, 1, 2}
91	90, 48	eleqtrri 2833	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
92		eqid 2733	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
93	84	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐷 = 𝐶 ↔ 𝐶 = 𝐶))
94	93	rspcev 3613	. . . . . . . . . 10 ⊢ ((2 ∈ (0..^3) ∧ 𝐶 = 𝐶) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
95	91, 92, 94	mp2an 691	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐶
96	95	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
97	88, 96	r19.29a 3163	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
98	97	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐶 ∈ ℂ)
99	85, 98	eqeltrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 ∈ ℂ)
100	83, 99	sylan2b 595	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {2}) → 𝐷 ∈ ℂ)
101	82, 100	fprodcl 15896	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 ∈ ℂ)
102	59, 80, 101	mulassd 11237	. 2 ⊢ (𝜑 → ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷) = (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)))
103		0nn0 12487	. . . . 5 ⊢ 0 ∈ ℕ₀
104	103	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
105	41	prodsn 15906	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ 𝐴 ∈ ℂ) → ∏𝑘 ∈ {0}𝐷 = 𝐴)
106	104, 55, 105	syl2anc 585	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 = 𝐴)
107		1nn0 12488	. . . . . 6 ⊢ 1 ∈ ℕ₀
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
109	63	prodsn 15906	. . . . 5 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {1}𝐷 = 𝐵)
110	108, 76, 109	syl2anc 585	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 = 𝐵)
111		2nn0 12489	. . . . . 6 ⊢ 2 ∈ ℕ₀
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
113	84	prodsn 15906	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {2}𝐷 = 𝐶)
114	112, 97, 113	syl2anc 585	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 = 𝐶)
115	110, 114	oveq12d 7427	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷) = (𝐵 · 𝐶))
116	106, 115	oveq12d 7427	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)) = (𝐴 · (𝐵 · 𝐶)))
117	37, 102, 116	3eqtrd 2777	1 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 {cpr 4631 {ctp 4633 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ≤ cle 11249 2c2 12267 3c3 12268 ℕ₀cn0 12472 ℤcz 12558 ℤ_≥cuz 12822 ..^cfzo 13627 ∏cprod 15849

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-prod 15850

This theorem is referenced by: circlevma 33654 circlemethhgt 33655

W3C validator