Theorem prod1dc 11549

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 2170	. . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simp1 992	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝑀 ∈ ℤ)
3		1ap0 8509	. . . 4 ⊢ 1 # 0
4	3	a1i 9	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 1 # 0)
5	1	prodfclim1 11507	. . . 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 994	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
8		eleq1w 2231	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑗 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
9	8	dcbid 833	. . . . 5 ⊢ (𝑗 = 𝑎 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑎 ∈ 𝐴))
10	9	cbvralv 2696	. . . 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 993	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝐴 ⊆ (ℤ≥‘𝑀))
13		1ex 7915	. . . . . 6 ⊢ 1 ∈ V
14	13	fvconst2 5712	. . . . 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 2231	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
17	16	dcbid 833	. . . . . 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 2839	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
21		ifiddc 3559	. . . . 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 2206	. . 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 7936	. . 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 11544	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
26		fz1f1o 11338	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
27		prodeq1 11516	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = ∏𝑘 ∈ ∅ 1)
28		prod0 11548	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 1 = 1
29	27, 28	eqtrdi 2219	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = 1)
30		eqidd 2171	. . . . . . . . . 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 7936	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
34		elfznn 10010	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
35	13	fvconst2 5712	. . . . . . . . . . . 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 11546	. . . . . . . . 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 3533	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = ((ℕ × {1})‘𝑗))
41	35	ad2antlr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
42	40, 41	eqtrd 2203	. . . . . . . . . . . . . . 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 3536	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
45		nnz 9231	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
46		nnz 9231	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℤ)
47		zdcle 9288	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑗 ≤ (♯‘𝐴))
48	45, 46, 47	syl2anr 288	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49		exmiddc 831	. . . . . . . . . . . . . . . 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 793	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
52	51	mpteq2dva 4079	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (𝑗 ∈ ℕ ↦ 1))
53		fconstmpt 4658	. . . . . . . . . . . . 13 ⊢ (ℕ × {1}) = (𝑗 ∈ ℕ ↦ 1)
54	52, 53	eqtr4di 2221	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
55	54	seqeq3d 10409	. . . . . . . . . . 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 5498	. . . . . . . . 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 2203	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59		nnuz 9522	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
60	59	prodf1 11505	. . . . . . . . 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 2203	. . . . . . 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 1812	. . . . 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 711	. . 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 711	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 703  DECID wdc 829   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121  ∅c0 3414  ifcif 3526  {csn 3583   class class class wbr 3989   ↦ cmpt 4050   × cxp 4609  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853  Fincfn 6718  0cc0 7774  1c1 7775   · cmul 7779   ≤ cle 7955   # cap 8500  ℕcn 8878  ℤcz 9212  ℤ_≥cuz 9487  ...cfz 9965  seqcseq 10401  ♯chash 10709   ⇝ cli 11241  ∏cprod 11513

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514

This theorem is referenced by: (None)