Theorem prod1dc 11768

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 2196	. . 3 ⊢ (ℤ≥‘𝑀) = (ℤ≥‘𝑀)
2		simp1 999	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝑀 ∈ ℤ)
3		1ap0 8634	. . . 4 ⊢ 1 # 0
4	3	a1i 9	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 1 # 0)
5	1	prodfclim1 11726	. . . 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 1001	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴)
8		eleq1w 2257	. . . . . 6 ⊢ (𝑗 = 𝑎 → (𝑗 ∈ 𝐴 ↔ 𝑎 ∈ 𝐴))
9	8	dcbid 839	. . . . 5 ⊢ (𝑗 = 𝑎 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑎 ∈ 𝐴))
10	9	cbvralv 2729	. . . 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 1000	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → 𝐴 ⊆ (ℤ≥‘𝑀))
13		1ex 8038	. . . . . 6 ⊢ 1 ∈ V
14	13	fvconst2 5781	. . . . 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 2257	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑎 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
17	16	dcbid 839	. . . . . 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 2873	. . . . 5 ⊢ (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → DECID 𝑘 ∈ 𝐴)
21		ifiddc 3596	. . . . 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 2232	. . 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 8059	. . 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 11763	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ≥‘𝑀) ∧ ∀𝑗 ∈ (ℤ≥‘𝑀)DECID 𝑗 ∈ 𝐴) → ∏𝑘 ∈ 𝐴 1 = 1)
26		fz1f1o 11557	. . 3 ⊢ (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)))
27		prodeq1 11735	. . . . 5 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = ∏𝑘 ∈ ∅ 1)
28		prod0 11767	. . . . 5 ⊢ ∏𝑘 ∈ ∅ 1 = 1
29	27, 28	eqtrdi 2245	. . . 4 ⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 1 = 1)
30		eqidd 2197	. . . . . . . . . 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 8059	. . . . . . . . . 10 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) ∧ 𝑘 ∈ 𝐴) → 1 ∈ ℂ)
34		elfznn 10146	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
35	13	fvconst2 5781	. . . . . . . . . . . 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 11765	. . . . . . . . 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 3569	. . . . . . . . . . . . . . . 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 2229	. . . . . . . . . . . . . . 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 3572	. . . . . . . . . . . . . . 15 ⊢ ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
45		nnz 9362	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
46		nnz 9362	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℤ)
47		zdcle 9419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑗 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑗 ≤ (♯‘𝐴))
48	45, 46, 47	syl2anr 290	. . . . . . . . . . . . . . . 16 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49		exmiddc 837	. . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
52	51	mpteq2dva 4124	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (𝑗 ∈ ℕ ↦ 1))
53		fconstmpt 4711	. . . . . . . . . . . . 13 ⊢ (ℕ × {1}) = (𝑗 ∈ ℕ ↦ 1)
54	52, 53	eqtr4di 2247	. . . . . . . . . . . 12 ⊢ ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
55	54	seqeq3d 10564	. . . . . . . . . . 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 5563	. . . . . . . . 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 2229	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴) → ∏𝑘 ∈ 𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59		nnuz 9654	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
60	59	prodf1 11724	. . . . . . . . 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 2229	. . . . . . 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 1833	. . . . 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 717	. . 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 717	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 709  DECID wdc 835   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ∅c0 3451  ifcif 3562  {csn 3623   class class class wbr 4034   ↦ cmpt 4095   × cxp 4662  –1-1-onto→wf1o 5258  ‘cfv 5259  (class class class)co 5925  Fincfn 6808  0cc0 7896  1c1 7897   · cmul 7901   ≤ cle 8079   # cap 8625  ℕcn 9007  ℤcz 9343  ℤ_≥cuz 9618  ...cfz 10100  seqcseq 10556  ♯chash 10884   ⇝ cli 11460  ∏cprod 11732

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-ihash 10885  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-proddc 11733

This theorem is referenced by: (None)