Theorem prod1dc 12276

Description: Any product of one over a valid set is one. (Contributed by Scott Fenton, 7-Dec-2017.) (Revised by Jim Kingdon, 5-Aug-2024.)

Assertion

Ref	Expression
prod1dc	⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘 ∈ 𝐴 1 = 1)

Distinct variable groups:   𝐴,𝑗,𝑘   𝑗,𝑀,𝑘

Proof of Theorem prod1dc

Dummy variables 𝑎 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2234	. . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simp1 1024	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝑀 ∈ ℤ)
3		1ap0 8866	. . . 4 ⊢ 1 # 0
4	3	a1i 9	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 1 # 0)
5	1	prodfclim1 12234	. . . 4 ⊢ (𝑀 ∈ ℤ → seq𝑀( · , ((ℤ≥‘𝑀) × {1})) ⇝ 1)
6	2, 5	syl 14	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → seq𝑀( · , ((ℤ≥‘𝑀) × {1})) ⇝ 1)
7		simp3 1026	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
8		eleq1w 2295	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑗 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
9	8	dcbid 846	. . . . 5 ⊢ (𝑗 = 𝑎 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑎 ∈ 𝐴))
10	9	cbvralv 2780	. . . 4 ⊢ (∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
11	7, 10	sylib 122	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
12		simp2 1025	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝐴 ⊆ (ℤ≥‘𝑀))
13		1ex 8271	. . . . . 6 ⊢ 1 ∈ V
14	13	fvconst2 5902	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {1})‘𝑘) = 1)
15	14	adantl 277	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {1})‘𝑘) = 1)
16		eleq1w 2295	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
17	16	dcbid 846	. . . . . 6 ⊢ (𝑎 = 𝑘 → (DECID 𝑎 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
18	11	adantr 276	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
19		simpr 110	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
20	17, 18, 19	rspcdva 2928	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
21		ifiddc 3660	. . . . 5 ⊢ (DECID 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, 1, 1) = 1)
22	20, 21	syl 14	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → if(𝑘 ∈ 𝐴, 1, 1) = 1)
23	15, 22	eqtr4d 2270	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {1})‘𝑘) = if(𝑘 ∈ 𝐴, 1, 1))
24		1cnd 8292	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
25	1, 2, 4, 6, 11, 12, 23, 24	zprodap0 12271	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
26		fz1f1o 12064	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
27		prodeq1 12243	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = ∏𝑘 ∈ ∅ 1)
28		prod0 12275	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 1 = 1
29	27, 28	eqtrdi 2283	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = 1)
30		eqidd 2235	. . . . . . . . . 10 ⊢ (𝑘 = (𝑓‘𝑗) → 1 = 1)
31		simpl 109	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ)
32		simpr 110	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
33		1cnd 8292	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
34		elfznn 10391	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
35	13	fvconst2 5902	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → ((ℕ × {1})‘𝑗) = 1)
36	34, 35	syl 14	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
37	36	adantl 277	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑗 ∈ (1...(♯‘𝐴))) → ((ℕ × {1})‘𝑗) = 1)
38	30, 31, 32, 33, 37	fprodseq 12273	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = (seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)))‘(♯‘𝐴)))
39		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → 𝑗 ≤ (♯‘𝐴))
40	39	iftrued 3631	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = ((ℕ × {1})‘𝑗))
41	35	ad2antlr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
42	40, 41	eqtrd 2267	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
43		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → ¬ 𝑗 ≤ (♯‘𝐴))
44	43	iffalsed 3634	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
45		nnz 9598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
46		nnz 9598	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℤ)
47		zdcle 9656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑗 ≤ (♯‘𝐴))
48	45, 46, 47	syl2anr 290	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49		exmiddc 844	. . . . . . . . . . . . . . . 16 ⊢ (DECID 𝑗 ≤ (♯‘𝐴) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
50	48, 49	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
51	42, 44, 50	mpjaodan 806	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
52	51	mpteq2dva 4202	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (𝑗 ∈ ℕ ↦ 1))
53		fconstmpt 4799	. . . . . . . . . . . . 13 ⊢ (ℕ × {1}) = (𝑗 ∈ ℕ ↦ 1)
54	52, 53	eqtr4di 2285	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
55	54	seqeq3d 10821	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
56	55	adantr 276	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
57	56	fveq1d 5674	. . . . . . . . 9 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)))‘(♯‘𝐴)) = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
58	38, 57	eqtrd 2267	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59		nnuz 9893	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
60	59	prodf1 12232	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
61	60	adantr 276	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
62	58, 61	eqtrd 2267	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
63	62	ex 115	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 1 = 1))
64	63	exlimdv 1868	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 1 = 1))
65	64	imp 124	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
66	29, 65	jaoi 724	. . 3 ⊢ ((𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)) → ∏𝑘 ∈ 𝐴 1 = 1)
67	26, 66	syl 14	. 2 ⊢ (𝐴 ∈ Fin → ∏𝑘 ∈ 𝐴 1 = 1)
68	25, 67	jaoi 724	1 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘 ∈ 𝐴 1 = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3213  ∅c0 3510  ifcif 3622  {csn 3691   class class class wbr 4111   ↦ cmpt 4173   × cxp 4749  –1-1-onto→wf1o 5353  ‘cfv 5354  (class class class)co 6052  Fincfn 6977  0cc0 8129  1c1 8130   · cmul 8134   ≤ cle 8311   # cap 8857  ℕcn 9239  ℤcz 9579  ℤ_≥cuz 9856  ...cfz 10345  seqcseq 10813  ♯chash 11142   ⇝ cli 11967  ∏cprod 12240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248  ax-caucvg 8249

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-isom 5363  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-n0 9499  df-z 9580  df-uz 9857  df-q 9955  df-rp 9990  df-fz 10346  df-fzo 10481  df-seqfrec 10814  df-exp 10905  df-ihash 11143  df-cj 11531  df-re 11532  df-im 11533  df-rsqrt 11687  df-abs 11688  df-clim 11968  df-proddc 12241

This theorem is referenced by: (None)