prodfzo03 - Mathbox for Thierry Arnoux

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Theorem prodfzo03 34637

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 13599	. . . . 5 ⊢ ((0..^2) ∩ {2}) = ∅
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ((0..^2) ∩ {2}) = ∅)
3		2p1e3 12269	. . . . . . 7 ⊢ (2 + 1) = 3
4	3	oveq2i 7363	. . . . . 6 ⊢ (0..^(2 + 1)) = (0..^3)
5		2eluzge0 12781	. . . . . . 7 ⊢ 2 ∈ (ℤ_≥‘0)
6		fzosplitsn 13678	. . . . . . 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 2758	. . . . 5 ⊢ (0..^3) = ((0..^2) ∪ {2})
9	8	a1i 11	. . . 4 ⊢ (𝜑 → (0..^3) = ((0..^2) ∪ {2}))
10		fzofi 13883	. . . . 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 15875	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷))
14		0ne1 12203	. . . . . 6 ⊢ 0 ≠ 1
15		disjsn2 4664	. . . . . 6 ⊢ (0 ≠ 1 → ({0} ∩ {1}) = ∅)
16	14, 15	mp1i 13	. . . . 5 ⊢ (𝜑 → ({0} ∩ {1}) = ∅)
17		fzo0to2pr 13652	. . . . . . 7 ⊢ (0..^2) = {0, 1}
18		df-pr 4578	. . . . . . 7 ⊢ {0, 1} = ({0} ∪ {1})
19	17, 18	eqtri 2756	. . . . . 6 ⊢ (0..^2) = ({0} ∪ {1})
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) = ({0} ∪ {1}))
21		fzofi 13883	. . . . . 6 ⊢ (0..^2) ∈ Fin
22	21	a1i 11	. . . . 5 ⊢ (𝜑 → (0..^2) ∈ Fin)
23		2z 12510	. . . . . . . . 9 ⊢ 2 ∈ ℤ
24		3z 12511	. . . . . . . . 9 ⊢ 3 ∈ ℤ
25		2re 12206	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
26		3re 12212	. . . . . . . . . 10 ⊢ 3 ∈ ℝ
27		2lt3 12299	. . . . . . . . . 10 ⊢ 2 < 3
28	25, 26, 27	ltleii 11243	. . . . . . . . 9 ⊢ 2 ≤ 3
29		eluz2 12744	. . . . . . . . 9 ⊢ (3 ∈ (ℤ_≥‘2) ↔ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3))
30	23, 24, 28, 29	mpbir3an 1342	. . . . . . . 8 ⊢ 3 ∈ (ℤ_≥‘2)
31		fzoss2 13589	. . . . . . . 8 ⊢ (3 ∈ (ℤ_≥‘2) → (0..^2) ⊆ (0..^3))
32	30, 31	ax-mp 5	. . . . . . 7 ⊢ (0..^2) ⊆ (0..^3)
33	32	sseli 3926	. . . . . 6 ⊢ (𝑘 ∈ (0..^2) → 𝑘 ∈ (0..^3))
34	33, 12	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^2)) → 𝐷 ∈ ℂ)
35	16, 20, 22, 34	fprodsplit 15875	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ (0..^2)𝐷 = (∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷))
36	35	oveq1d 7367	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ (0..^2)𝐷 · ∏𝑘 ∈ {2}𝐷) = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
37	13, 36	eqtrd 2768	. 2 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷))
38		snfi 8972	. . . . 5 ⊢ {0} ∈ Fin
39	38	a1i 11	. . . 4 ⊢ (𝜑 → {0} ∈ Fin)
40		velsn 4591	. . . . 5 ⊢ (𝑘 ∈ {0} ↔ 𝑘 = 0)
41		prodfzo03.1	. . . . . . 7 ⊢ (𝑘 = 0 → 𝐷 = 𝐴)
42	41	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 = 𝐴)
43		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 = 𝐴)
44	12	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐷 ∈ ℂ)
45	43, 44	eqeltrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐴) → 𝐴 ∈ ℂ)
46		c0ex 11113	. . . . . . . . . . . 12 ⊢ 0 ∈ V
47	46	tpid1 4720	. . . . . . . . . . 11 ⊢ 0 ∈ {0, 1, 2}
48		fzo0to3tp 13654	. . . . . . . . . . 11 ⊢ (0..^3) = {0, 1, 2}
49	47, 48	eleqtrri 2832	. . . . . . . . . 10 ⊢ 0 ∈ (0..^3)
50		eqid 2733	. . . . . . . . . 10 ⊢ 𝐴 = 𝐴
51	41	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 0 → (𝐷 = 𝐴 ↔ 𝐴 = 𝐴))
52	51	rspcev 3573	. . . . . . . . . 10 ⊢ ((0 ∈ (0..^3) ∧ 𝐴 = 𝐴) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
53	49, 50, 52	mp2an 692	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐴
54	53	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐴)
55	45, 54	r19.29a 3141	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
56	55	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐴 ∈ ℂ)
57	42, 56	eqeltrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 0) → 𝐷 ∈ ℂ)
58	40, 57	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {0}) → 𝐷 ∈ ℂ)
59	39, 58	fprodcl 15861	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 ∈ ℂ)
60		snfi 8972	. . . . 5 ⊢ {1} ∈ Fin
61	60	a1i 11	. . . 4 ⊢ (𝜑 → {1} ∈ Fin)
62		velsn 4591	. . . . 5 ⊢ (𝑘 ∈ {1} ↔ 𝑘 = 1)
63		prodfzo03.2	. . . . . . 7 ⊢ (𝑘 = 1 → 𝐷 = 𝐵)
64	63	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 = 𝐵)
65		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 = 𝐵)
66	12	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐷 ∈ ℂ)
67	65, 66	eqeltrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐵) → 𝐵 ∈ ℂ)
68		1ex 11115	. . . . . . . . . . . 12 ⊢ 1 ∈ V
69	68	tpid2 4722	. . . . . . . . . . 11 ⊢ 1 ∈ {0, 1, 2}
70	69, 48	eleqtrri 2832	. . . . . . . . . 10 ⊢ 1 ∈ (0..^3)
71		eqid 2733	. . . . . . . . . 10 ⊢ 𝐵 = 𝐵
72	63	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 1 → (𝐷 = 𝐵 ↔ 𝐵 = 𝐵))
73	72	rspcev 3573	. . . . . . . . . 10 ⊢ ((1 ∈ (0..^3) ∧ 𝐵 = 𝐵) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
74	70, 71, 73	mp2an 692	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐵
75	74	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐵)
76	67, 75	r19.29a 3141	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
77	76	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐵 ∈ ℂ)
78	64, 77	eqeltrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
79	62, 78	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {1}) → 𝐷 ∈ ℂ)
80	61, 79	fprodcl 15861	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 ∈ ℂ)
81		snfi 8972	. . . . 5 ⊢ {2} ∈ Fin
82	81	a1i 11	. . . 4 ⊢ (𝜑 → {2} ∈ Fin)
83		velsn 4591	. . . . 5 ⊢ (𝑘 ∈ {2} ↔ 𝑘 = 2)
84		prodfzo03.3	. . . . . . 7 ⊢ (𝑘 = 2 → 𝐷 = 𝐶)
85	84	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 = 𝐶)
86		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 = 𝐶)
87	12	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐷 ∈ ℂ)
88	86, 87	eqeltrrd 2834	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^3)) ∧ 𝐷 = 𝐶) → 𝐶 ∈ ℂ)
89		2ex 12209	. . . . . . . . . . . 12 ⊢ 2 ∈ V
90	89	tpid3 4725	. . . . . . . . . . 11 ⊢ 2 ∈ {0, 1, 2}
91	90, 48	eleqtrri 2832	. . . . . . . . . 10 ⊢ 2 ∈ (0..^3)
92		eqid 2733	. . . . . . . . . 10 ⊢ 𝐶 = 𝐶
93	84	eqeq1d 2735	. . . . . . . . . . 11 ⊢ (𝑘 = 2 → (𝐷 = 𝐶 ↔ 𝐶 = 𝐶))
94	93	rspcev 3573	. . . . . . . . . 10 ⊢ ((2 ∈ (0..^3) ∧ 𝐶 = 𝐶) → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
95	91, 92, 94	mp2an 692	. . . . . . . . 9 ⊢ ∃𝑘 ∈ (0..^3)𝐷 = 𝐶
96	95	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ∃𝑘 ∈ (0..^3)𝐷 = 𝐶)
97	88, 96	r19.29a 3141	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℂ)
98	97	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐶 ∈ ℂ)
99	85, 98	eqeltrd 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 = 2) → 𝐷 ∈ ℂ)
100	83, 99	sylan2b 594	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ {2}) → 𝐷 ∈ ℂ)
101	82, 100	fprodcl 15861	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 ∈ ℂ)
102	59, 80, 101	mulassd 11142	. 2 ⊢ (𝜑 → ((∏𝑘 ∈ {0}𝐷 · ∏𝑘 ∈ {1}𝐷) · ∏𝑘 ∈ {2}𝐷) = (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)))
103		0nn0 12403	. . . . 5 ⊢ 0 ∈ ℕ₀
104	103	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℕ₀)
105	41	prodsn 15871	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ 𝐴 ∈ ℂ) → ∏𝑘 ∈ {0}𝐷 = 𝐴)
106	104, 55, 105	syl2anc 584	. . 3 ⊢ (𝜑 → ∏𝑘 ∈ {0}𝐷 = 𝐴)
107		1nn0 12404	. . . . . 6 ⊢ 1 ∈ ℕ₀
108	107	a1i 11	. . . . 5 ⊢ (𝜑 → 1 ∈ ℕ₀)
109	63	prodsn 15871	. . . . 5 ⊢ ((1 ∈ ℕ₀ ∧ 𝐵 ∈ ℂ) → ∏𝑘 ∈ {1}𝐷 = 𝐵)
110	108, 76, 109	syl2anc 584	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {1}𝐷 = 𝐵)
111		2nn0 12405	. . . . . 6 ⊢ 2 ∈ ℕ₀
112	111	a1i 11	. . . . 5 ⊢ (𝜑 → 2 ∈ ℕ₀)
113	84	prodsn 15871	. . . . 5 ⊢ ((2 ∈ ℕ₀ ∧ 𝐶 ∈ ℂ) → ∏𝑘 ∈ {2}𝐷 = 𝐶)
114	112, 97, 113	syl2anc 584	. . . 4 ⊢ (𝜑 → ∏𝑘 ∈ {2}𝐷 = 𝐶)
115	110, 114	oveq12d 7370	. . 3 ⊢ (𝜑 → (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷) = (𝐵 · 𝐶))
116	106, 115	oveq12d 7370	. 2 ⊢ (𝜑 → (∏𝑘 ∈ {0}𝐷 · (∏𝑘 ∈ {1}𝐷 · ∏𝑘 ∈ {2}𝐷)) = (𝐴 · (𝐵 · 𝐶)))
117	37, 102, 116	3eqtrd 2772	1 ⊢ (𝜑 → ∏𝑘 ∈ (0..^3)𝐷 = (𝐴 · (𝐵 · 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∃wrex 3057 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 {csn 4575 {cpr 4577 {ctp 4579 class class class wbr 5093 ‘cfv 6486 (class class class)co 7352 Fincfn 8875 ℂcc 11011 0cc0 11013 1c1 11014 + caddc 11016 · cmul 11018 ≤ cle 11154 2c2 12187 3c3 12188 ℕ₀cn0 12388 ℤcz 12475 ℤ_≥cuz 12738 ..^cfzo 13556 ∏cprod 15812

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-n0 12389 df-z 12476 df-uz 12739 df-rp 12893 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-prod 15813

This theorem is referenced by: circlevma 34676 circlemethhgt 34677

W3C validator