Theorem prod1dc 11527

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 2165	. . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simp1 987	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝑀 ∈ ℤ)
3		1ap0 8488	. . . 4 ⊢ 1 # 0
4	3	a1i 9	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 1 # 0)
5	1	prodfclim1 11485	. . . 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 989	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
8		eleq1w 2227	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑗 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
9	8	dcbid 828	. . . . 5 ⊢ (𝑗 = 𝑎 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑎 ∈ 𝐴))
10	9	cbvralv 2692	. . . 4 ⊢ (∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
11	7, 10	sylib 121	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
12		simp2 988	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝐴 ⊆ (ℤ≥‘𝑀))
13		1ex 7894	. . . . . 6 ⊢ 1 ∈ V
14	13	fvconst2 5701	. . . . 5 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {1})‘𝑘) = 1)
15	14	adantl 275	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (((ℤ≥‘𝑀) × {1})‘𝑘) = 1)
16		eleq1w 2227	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
17	16	dcbid 828	. . . . . 6 ⊢ (𝑎 = 𝑘 → (DECID 𝑎 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
18	11	adantr 274	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → ∀𝑎 ∈ (ℤ≥‘𝑀)DECID 𝑎 ∈ 𝐴)
19		simpr 109	. . . . . 6 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → 𝑘 ∈ (ℤ≥‘𝑀))
20	17, 18, 19	rspcdva 2835	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
21		ifiddc 3553	. . . . 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 2201	. . 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 7915	. . 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 11522	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
26		fz1f1o 11316	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
27		prodeq1 11494	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = ∏𝑘 ∈ ∅ 1)
28		prod0 11526	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 1 = 1
29	27, 28	eqtrdi 2215	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = 1)
30		eqidd 2166	. . . . . . . . . 10 ⊢ (𝑘 = (𝑓‘𝑗) → 1 = 1)
31		simpl 108	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (♯‘𝐴) ∈ ℕ)
32		simpr 109	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
33		1cnd 7915	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
34		elfznn 9989	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
35	13	fvconst2 5701	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ ℕ → ((ℕ × {1})‘𝑗) = 1)
36	34, 35	syl 14	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
37	36	adantl 275	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑗 ∈ (1...(♯‘𝐴))) → ((ℕ × {1})‘𝑗) = 1)
38	30, 31, 32, 33, 37	fprodseq 11524	. . . . . . . . 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 109	. . . . . . . . . . . . . . . . 17 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → 𝑗 ≤ (♯‘𝐴))
40	39	iftrued 3527	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = ((ℕ × {1})‘𝑗))
41	35	ad2antlr 481	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
42	40, 41	eqtrd 2198	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
43		simpr 109	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → ¬ 𝑗 ≤ (♯‘𝐴))
44	43	iffalsed 3530	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
45		nnz 9210	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
46		nnz 9210	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℤ)
47		zdcle 9267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑗 ≤ (♯‘𝐴))
48	45, 46, 47	syl2anr 288	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49		exmiddc 826	. . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
52	51	mpteq2dva 4072	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (𝑗 ∈ ℕ ↦ 1))
53		fconstmpt 4651	. . . . . . . . . . . . 13 ⊢ (ℕ × {1}) = (𝑗 ∈ ℕ ↦ 1)
54	52, 53	eqtr4di 2217	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
55	54	seqeq3d 10388	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℕ → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
56	55	adantr 274	. . . . . . . . . 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 5488	. . . . . . . . 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 2198	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59		nnuz 9501	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
60	59	prodf1 11483	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
61	60	adantr 274	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
62	58, 61	eqtrd 2198	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
63	62	ex 114	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 1 = 1))
64	63	exlimdv 1807	. . . . 5 ⊢ ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴 → ∏𝑘 ∈ 𝐴 1 = 1))
65	64	imp 123	. . . 4 ⊢ (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
66	29, 65	jaoi 706	. . 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 706	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 103   ∨ wo 698  DECID wdc 824   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ∅c0 3409  ifcif 3520  {csn 3576   class class class wbr 3982   ↦ cmpt 4043   × cxp 4602  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842  Fincfn 6706  0cc0 7753  1c1 7754   · cmul 7758   ≤ cle 7934   # cap 8479  ℕcn 8857  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  seqcseq 10380  ♯chash 10688   ⇝ cli 11219  ∏cprod 11491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492

This theorem is referenced by: (None)