prodfzo03 - Mathbox for Thierry Arnoux

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Theorem prodfzo03 34971

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 13728	. . . . 5 ⊢ ((0..^2) ∩ {2}) = ∅
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ((0..^2) ∩ {2}) = ∅)
3		2p1e3 12383	. . . . . . 7 ⊢ (2 + 1) = 3
4	3	oveq2i 7423	. . . . . 6 ⊢ (0..^(2 + 1)) = (0..^3)
5		2eluzge0 12906	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
6		fzosplitsn 13807	. . . . . . 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 2788	. . . . 5 ⊢ (0..^3) = ((0..^2) ∪ {2})
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) = ((0..^2) ∪ {2}))
10		fzofi 14012	. . . . 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 16022	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷))
14		0ne1 12313	. . . . . 6 ⊢ 0 ≠ 1
15		disjsn2 4679	. . . . . 6 ⊢ (0 ≠ 1 → ({0} ∩ {1}) = ∅)
16	14, 15	mp1i 14	. . . . 5 ⊢ (𝜑 → ({0} ∩ {1}) = ∅)
17		fzo0to2pr 13781	. . . . . . 7 ⊢ (0..^2) = {0, 1}
18		df-pr 4593	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
19	17, 18	eqtri 2786	. . . . . 6 ⊢ (0..^2) = ({0} ∪ {1})
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) = ({0} ∪ {1}))
21		fzofi 14012	. . . . . 6 ⊢ (0..^2) ∈ Fin
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) ∈ Fin)
23		2z 12627	. . . . . . . . 9 ⊢ 2 ∈ ℤ
24		3z 12628	. . . . . . . . 9 ⊢ 3 ∈ ℤ
25		2re 12316	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
26		3re 12322	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
27		2lt3 12415	. . . . . . . . . 10 ⊢ 2 < 3
28	25, 26, 27	ltleii 11334	. . . . . . . . 9 ⊢ 2 ≤ 3
29		eluz2 12869	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3))
30	23, 24, 28, 29	mpbir3an 1360	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
31		fzoss2 13718	. . . . . . . 8 ⊢ (3 ∈ (ℤ_≥‘2) → (0..^2) ⊆ (0..^3))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (0..^2) ⊆ (0..^3)
33	32	sseli 3934	. . . . . 6 ⊢ (𝑘 ∈ (0..^2) → 𝑘 ∈ (0..^3))
34	33, 12	sylan2 604	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^2)) → 𝐷 ∈ ℂ)
35	16, 20, 22, 34	fprodsplit 16022	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (0..^2)𝐷 = (∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷))
36	35	oveq1d 7427	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷) = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
37	13, 36	eqtrd 2798	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
38		snfi 9041	. . . . 5 ⊢ {0} ∈ Fin
39	38	a1i 11	. . . 4 ⊢ (𝜑 → {0} ∈ Fin)
40		velsn 4606	. . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
41		prodfzo03.1	. . . . . . 7 ⊢ (𝑘 = 0 → 𝐷 = 𝐴)
42	41	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 = 𝐴)
43		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 = 𝐴)
44	12	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 ∈ ℂ)
45	43, 44	eqeltrrd 2864	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐴 ∈ ℂ)
46		c0ex 11201	. . . . . . . . . . . 12 ⊢ 0 ∈ V
47	46	tpid1 4735	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1, 2}
48		fzo0to3tp 13783	. . . . . . . . . . 11 ⊢ (0..^3) = {0, 1, 2}
49	47, 48	eleqtrri 2862	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
50		eqid 2763	. . . . . . . . . 10 ⊢ 𝐴 = 𝐴
51	41	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝐷 = 𝐴 ↔ 𝐴 = 𝐴))
52	51	rspcev 3582	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^3) ∧ 𝐴 = 𝐴) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
53	49, 50, 52	mp2an 704	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐴
54	53	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
55	45, 54	r19.29a 3173	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
56	55	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 ∈ ℂ)
57	42, 56	eqeltrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 ∈ ℂ)
58	40, 57	sylan2b 605	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝐷 ∈ ℂ)
59	39, 58	fprodcl 16008	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 ∈ ℂ)
60		snfi 9041	. . . . 5 ⊢ {1} ∈ Fin
61	60	a1i 11	. . . 4 ⊢ (𝜑 → {1} ∈ Fin)
62		velsn 4606	. . . . 5 ⊢ (𝑘 ∈ {1} ↔ 𝑘 = 1)
63		prodfzo03.2	. . . . . . 7 ⊢ (𝑘 = 1 → 𝐷 = 𝐵)
64	63	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 = 𝐵)
65		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 = 𝐵)
66	12	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 ∈ ℂ)
67	65, 66	eqeltrrd 2864	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐵 ∈ ℂ)
68		1eltp012 12312	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1, 2}
69	68, 48	eleqtrri 2862	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
70		eqid 2763	. . . . . . . . . 10 ⊢ 𝐵 = 𝐵
71	63	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐷 = 𝐵 ↔ 𝐵 = 𝐵))
72	71	rspcev 3582	. . . . . . . . . 10 ⊢ ((1 ∈ (0..^3) ∧ 𝐵 = 𝐵) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
73	69, 70, 72	mp2an 704	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐵
74	73	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
75	67, 74	r19.29a 3173	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
76	75	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐵 ∈ ℂ)
77	64, 76	eqeltrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
78	62, 77	sylan2b 605	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {1}) → 𝐷 ∈ ℂ)
79	61, 78	fprodcl 16008	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 ∈ ℂ)
80		snfi 9041	. . . . 5 ⊢ {2} ∈ Fin
81	80	a1i 11	. . . 4 ⊢ (𝜑 → {2} ∈ Fin)
82		velsn 4606	. . . . 5 ⊢ (𝑘 ∈ {2} ↔ 𝑘 = 2)
83		prodfzo03.3	. . . . . . 7 ⊢ (𝑘 = 2 → 𝐷 = 𝐶)
84	83	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 = 𝐶)
85		simpr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 = 𝐶)
86	12	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 ∈ ℂ)
87	85, 86	eqeltrrd 2864	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐶 ∈ ℂ)
88		2ex 12319	. . . . . . . . . . . 12 ⊢ 2 ∈ V
89	88	tpid3 4740	. . . . . . . . . . 11 ⊢ 2 ∈ {0, 1, 2}
90	89, 48	eleqtrri 2862	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
91		eqid 2763	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
92	83	eqeq1d 2765	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐷 = 𝐶 ↔ 𝐶 = 𝐶))
93	92	rspcev 3582	. . . . . . . . . 10 ⊢ ((2 ∈ (0..^3) ∧ 𝐶 = 𝐶) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
94	90, 91, 93	mp2an 704	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐶
95	94	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
96	87, 95	r19.29a 3173	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
97	96	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐶 ∈ ℂ)
98	84, 97	eqeltrd 2863	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 ∈ ℂ)
99	82, 98	sylan2b 605	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {2}) → 𝐷 ∈ ℂ)
100	81, 99	fprodcl 16008	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 ∈ ℂ)
101	59, 79, 100	mulassd 11233	. 2 ⊢ (𝜑 → ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷) = (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)))
102		0nn0 12520	. . . . 5 ⊢ 0 ∈ ℕ₀
103	102	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
104	41	prodsn 16018	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ 𝐴 ∈ ℂ) → ∏𝑘 ∈ {0}𝐷 = 𝐴)
105	103, 55, 104	syl2anc 595	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 = 𝐴)
106		1nn0 12521	. . . . . 6 ⊢ 1 ∈ ℕ₀
107	106	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
108	63	prodsn 16018	. . . . 5 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {1}𝐷 = 𝐵)
109	107, 75, 108	syl2anc 595	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 = 𝐵)
110		2nn0 12522	. . . . . 6 ⊢ 2 ∈ ℕ₀
111	110	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
112	83	prodsn 16018	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {2}𝐷 = 𝐶)
113	111, 96, 112	syl2anc 595	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 = 𝐶)
114	109, 113	oveq12d 7430	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷) = (𝐵 · 𝐶))
115	105, 114	oveq12d 7430	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)) = (𝐴 · (𝐵 · 𝐶)))
116	37, 101, 115	3eqtrd 2802	1 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {csn 4590 {cpr 4592 {ctp 4594 class class class wbr 5110 ‘cfv 6538 (class class class)co 7412 Fincfn 8944 ℂcc 11099 0cc0 11101 1c1 11102 + caddc 11104 · cmul 11106 ≤ cle 11245 2c2 12296 3c3 12297 ℕ₀cn0 12505 ℤcz 12592 ℤ_≥cuz 12863 ..^cfzo 13684 ∏cprod 15959

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-prod 15960

This theorem is referenced by: circlevma 35010 circlemethhgt 35011

W3C validator