prodfzo03 - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > prodfzo03		Structured version Visualization version GIF version

Theorem prodfzo03 34861

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 13700	. . . . 5 ⊢ ((0..^2) ∩ {2}) = ∅
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ((0..^2) ∩ {2}) = ∅)
3		2p1e3 12356	. . . . . . 7 ⊢ (2 + 1) = 3
4	3	oveq2i 7403	. . . . . 6 ⊢ (0..^(2 + 1)) = (0..^3)
5		2eluzge0 12879	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
6		fzosplitsn 13779	. . . . . . 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 2786	. . . . 5 ⊢ (0..^3) = ((0..^2) ∪ {2})
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) = ((0..^2) ∪ {2}))
10		fzofi 13984	. . . . 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 15979	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷))
14		0ne1 12286	. . . . . 6 ⊢ 0 ≠ 1
15		disjsn2 4670	. . . . . 6 ⊢ (0 ≠ 1 → ({0} ∩ {1}) = ∅)
16	14, 15	mp1i 13	. . . . 5 ⊢ (𝜑 → ({0} ∩ {1}) = ∅)
17		fzo0to2pr 13753	. . . . . . 7 ⊢ (0..^2) = {0, 1}
18		df-pr 4584	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
19	17, 18	eqtri 2784	. . . . . 6 ⊢ (0..^2) = ({0} ∪ {1})
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) = ({0} ∪ {1}))
21		fzofi 13984	. . . . . 6 ⊢ (0..^2) ∈ Fin
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) ∈ Fin)
23		2z 12600	. . . . . . . . 9 ⊢ 2 ∈ ℤ
24		3z 12601	. . . . . . . . 9 ⊢ 3 ∈ ℤ
25		2re 12289	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
26		3re 12295	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
27		2lt3 12388	. . . . . . . . . 10 ⊢ 2 < 3
28	25, 26, 27	ltleii 11303	. . . . . . . . 9 ⊢ 2 ≤ 3
29		eluz2 12842	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3))
30	23, 24, 28, 29	mpbir3an 1354	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
31		fzoss2 13690	. . . . . . . 8 ⊢ (3 ∈ (ℤ_≥‘2) → (0..^2) ⊆ (0..^3))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (0..^2) ⊆ (0..^3)
33	32	sseli 3932	. . . . . 6 ⊢ (𝑘 ∈ (0..^2) → 𝑘 ∈ (0..^3))
34	33, 12	sylan2 602	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^2)) → 𝐷 ∈ ℂ)
35	16, 20, 22, 34	fprodsplit 15979	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (0..^2)𝐷 = (∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷))
36	35	oveq1d 7407	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷) = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
37	13, 36	eqtrd 2796	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
38		snfi 9020	. . . . 5 ⊢ {0} ∈ Fin
39	38	a1i 11	. . . 4 ⊢ (𝜑 → {0} ∈ Fin)
40		velsn 4597	. . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
41		prodfzo03.1	. . . . . . 7 ⊢ (𝑘 = 0 → 𝐷 = 𝐴)
42	41	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 = 𝐴)
43		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 = 𝐴)
44	12	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 ∈ ℂ)
45	43, 44	eqeltrrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐴 ∈ ℂ)
46		c0ex 11170	. . . . . . . . . . . 12 ⊢ 0 ∈ V
47	46	tpid1 4726	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1, 2}
48		fzo0to3tp 13755	. . . . . . . . . . 11 ⊢ (0..^3) = {0, 1, 2}
49	47, 48	eleqtrri 2860	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
50		eqid 2761	. . . . . . . . . 10 ⊢ 𝐴 = 𝐴
51	41	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝐷 = 𝐴 ↔ 𝐴 = 𝐴))
52	51	rspcev 3581	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^3) ∧ 𝐴 = 𝐴) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
53	49, 50, 52	mp2an 702	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐴
54	53	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
55	45, 54	r19.29a 3169	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
56	55	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 ∈ ℂ)
57	42, 56	eqeltrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 ∈ ℂ)
58	40, 57	sylan2b 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝐷 ∈ ℂ)
59	39, 58	fprodcl 15965	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 ∈ ℂ)
60		snfi 9020	. . . . 5 ⊢ {1} ∈ Fin
61	60	a1i 11	. . . 4 ⊢ (𝜑 → {1} ∈ Fin)
62		velsn 4597	. . . . 5 ⊢ (𝑘 ∈ {1} ↔ 𝑘 = 1)
63		prodfzo03.2	. . . . . . 7 ⊢ (𝑘 = 1 → 𝐷 = 𝐵)
64	63	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 = 𝐵)
65		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 = 𝐵)
66	12	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 ∈ ℂ)
67	65, 66	eqeltrrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐵 ∈ ℂ)
68		1ex 11173	. . . . . . . . . . . 12 ⊢ 1 ∈ V
69	68	tpid2 4728	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1, 2}
70	69, 48	eleqtrri 2860	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
71		eqid 2761	. . . . . . . . . 10 ⊢ 𝐵 = 𝐵
72	63	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐷 = 𝐵 ↔ 𝐵 = 𝐵))
73	72	rspcev 3581	. . . . . . . . . 10 ⊢ ((1 ∈ (0..^3) ∧ 𝐵 = 𝐵) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
74	70, 71, 73	mp2an 702	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐵
75	74	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
76	67, 75	r19.29a 3169	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
77	76	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐵 ∈ ℂ)
78	64, 77	eqeltrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
79	62, 78	sylan2b 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {1}) → 𝐷 ∈ ℂ)
80	61, 79	fprodcl 15965	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 ∈ ℂ)
81		snfi 9020	. . . . 5 ⊢ {2} ∈ Fin
82	81	a1i 11	. . . 4 ⊢ (𝜑 → {2} ∈ Fin)
83		velsn 4597	. . . . 5 ⊢ (𝑘 ∈ {2} ↔ 𝑘 = 2)
84		prodfzo03.3	. . . . . . 7 ⊢ (𝑘 = 2 → 𝐷 = 𝐶)
85	84	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 = 𝐶)
86		simpr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 = 𝐶)
87	12	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 ∈ ℂ)
88	86, 87	eqeltrrd 2862	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐶 ∈ ℂ)
89		2ex 12292	. . . . . . . . . . . 12 ⊢ 2 ∈ V
90	89	tpid3 4731	. . . . . . . . . . 11 ⊢ 2 ∈ {0, 1, 2}
91	90, 48	eleqtrri 2860	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
92		eqid 2761	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
93	84	eqeq1d 2763	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐷 = 𝐶 ↔ 𝐶 = 𝐶))
94	93	rspcev 3581	. . . . . . . . . 10 ⊢ ((2 ∈ (0..^3) ∧ 𝐶 = 𝐶) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
95	91, 92, 94	mp2an 702	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐶
96	95	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
97	88, 96	r19.29a 3169	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
98	97	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐶 ∈ ℂ)
99	85, 98	eqeltrd 2861	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 ∈ ℂ)
100	83, 99	sylan2b 603	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {2}) → 𝐷 ∈ ℂ)
101	82, 100	fprodcl 15965	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 ∈ ℂ)
102	59, 80, 101	mulassd 11202	. 2 ⊢ (𝜑 → ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷) = (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)))
103		0nn0 12493	. . . . 5 ⊢ 0 ∈ ℕ₀
104	103	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
105	41	prodsn 15975	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ 𝐴 ∈ ℂ) → ∏𝑘 ∈ {0}𝐷 = 𝐴)
106	104, 55, 105	syl2anc 593	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 = 𝐴)
107		1nn0 12494	. . . . . 6 ⊢ 1 ∈ ℕ₀
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
109	63	prodsn 15975	. . . . 5 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {1}𝐷 = 𝐵)
110	108, 76, 109	syl2anc 593	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 = 𝐵)
111		2nn0 12495	. . . . . 6 ⊢ 2 ∈ ℕ₀
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
113	84	prodsn 15975	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {2}𝐷 = 𝐶)
114	112, 97, 113	syl2anc 593	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 = 𝐶)
115	110, 114	oveq12d 7410	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷) = (𝐵 · 𝐶))
116	106, 115	oveq12d 7410	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)) = (𝐴 · (𝐵 · 𝐶)))
117	37, 102, 116	3eqtrd 2800	1 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 ∪ cun 3902 ∩ cin 3903 ⊆ wss 3904 ∅c0 4285 {csn 4581 {cpr 4583 {ctp 4585 class class class wbr 5099 ‘cfv 6517 (class class class)co 7392 Fincfn 8923 ℂcc 11068 0cc0 11070 1c1 11071 + caddc 11073 · cmul 11075 ≤ cle 11214 2c2 12269 3c3 12270 ℕ₀cn0 12478 ℤcz 12565 ℤ_≥cuz 12836 ..^cfzo 13656 ∏cprod 15916

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-oi 9455 df-card 9894 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-rp 12991 df-fz 13510 df-fzo 13657 df-seq 14012 df-exp 14072 df-hash 14341 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-clim 15498 df-prod 15917

This theorem is referenced by: circlevma 34900 circlemethhgt 34901

W3C validator