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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 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘𝐴 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.
StepHypRef Expression
1 eqid 2170 . . 3 (ℤ𝑀) = (ℤ𝑀)
2 simp1 992 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 𝑀 ∈ ℤ)
3 1ap0 8509 . . . 4 1 # 0
43a1i 9 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 1 # 0)
51prodfclim1 11507 . . . 4 (𝑀 ∈ ℤ → seq𝑀( · , ((ℤ𝑀) × {1})) ⇝ 1)
62, 5syl 14 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → seq𝑀( · , ((ℤ𝑀) × {1})) ⇝ 1)
7 simp3 994 . . . 4 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
8 eleq1w 2231 . . . . . 6 (𝑗 = 𝑎 → (𝑗𝐴𝑎𝐴))
98dcbid 833 . . . . 5 (𝑗 = 𝑎 → (DECID 𝑗𝐴DECID 𝑎𝐴))
109cbvralv 2696 . . . 4 (∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ↔ ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
117, 10sylib 121 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
12 simp2 993 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 𝐴 ⊆ (ℤ𝑀))
13 1ex 7915 . . . . . 6 1 ∈ V
1413fvconst2 5712 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (((ℤ𝑀) × {1})‘𝑘) = 1)
1514adantl 275 . . . 4 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → (((ℤ𝑀) × {1})‘𝑘) = 1)
16 eleq1w 2231 . . . . . . 7 (𝑎 = 𝑘 → (𝑎𝐴𝑘𝐴))
1716dcbid 833 . . . . . 6 (𝑎 = 𝑘 → (DECID 𝑎𝐴DECID 𝑘𝐴))
1811adantr 274 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
19 simpr 109 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
2017, 18, 19rspcdva 2839 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)
21 ifiddc 3559 . . . . 5 (DECID 𝑘𝐴 → if(𝑘𝐴, 1, 1) = 1)
2220, 21syl 14 . . . 4 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → if(𝑘𝐴, 1, 1) = 1)
2315, 22eqtr4d 2206 . . 3 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → (((ℤ𝑀) × {1})‘𝑘) = if(𝑘𝐴, 1, 1))
24 1cnd 7936 . . 3 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘𝐴) → 1 ∈ ℂ)
251, 2, 4, 6, 11, 12, 23, 24zprodap0 11544 . 2 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∏𝑘𝐴 1 = 1)
26 fz1f1o 11338 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
27 prodeq1 11516 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 1 = ∏𝑘 ∈ ∅ 1)
28 prod0 11548 . . . . 5 𝑘 ∈ ∅ 1 = 1
2927, 28eqtrdi 2219 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 1 = 1)
30 eqidd 2171 . . . . . . . . . 10 (𝑘 = (𝑓𝑗) → 1 = 1)
31 simpl 108 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (♯‘𝐴) ∈ ℕ)
32 simpr 109 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
33 1cnd 7936 . . . . . . . . . 10 ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ 𝑘𝐴) → 1 ∈ ℂ)
34 elfznn 10010 . . . . . . . . . . . 12 (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
3513fvconst2 5712 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → ((ℕ × {1})‘𝑗) = 1)
3634, 35syl 14 . . . . . . . . . . 11 (𝑗 ∈ (1...(♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
3736adantl 275 . . . . . . . . . 10 ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ 𝑗 ∈ (1...(♯‘𝐴))) → ((ℕ × {1})‘𝑗) = 1)
3830, 31, 32, 33, 37fprodseq 11546 . . . . . . . . 9 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = (seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)))‘(♯‘𝐴)))
39 simpr 109 . . . . . . . . . . . . . . . . 17 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → 𝑗 ≤ (♯‘𝐴))
4039iftrued 3533 . . . . . . . . . . . . . . . 16 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = ((ℕ × {1})‘𝑗))
4135ad2antlr 486 . . . . . . . . . . . . . . . 16 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
4240, 41eqtrd 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))
4443iffalsed 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))
4845, 46, 47syl2anr 288 . . . . . . . . . . . . . . . 16 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49 exmiddc 831 . . . . . . . . . . . . . . . 16 (DECID 𝑗 ≤ (♯‘𝐴) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
5048, 49syl 14 . . . . . . . . . . . . . . 15 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
5142, 44, 50mpjaodan 793 . . . . . . . . . . . . . 14 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
5251mpteq2dva 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)
5452, 53eqtr4di 2221 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
5554seqeq3d 10409 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
5655adantr 274 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
5756fveq1d 5498 . . . . . . . . 9 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)))‘(♯‘𝐴)) = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
5838, 57eqtrd 2203 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59 nnuz 9522 . . . . . . . . . 10 ℕ = (ℤ‘1)
6059prodf1 11505 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
6160adantr 274 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
6258, 61eqtrd 2203 . . . . . . 7 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = 1)
6362ex 114 . . . . . 6 ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 1 = 1))
6463exlimdv 1812 . . . . 5 ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 1 = 1))
6564imp 123 . . . 4 (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = 1)
6629, 65jaoi 711 . . 3 ((𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)) → ∏𝑘𝐴 1 = 1)
6726, 66syl 14 . 2 (𝐴 ∈ Fin → ∏𝑘𝐴 1 = 1)
6825, 67jaoi 711 1 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∨ 𝐴 ∈ Fin) → ∏𝑘𝐴 1 = 1)
Colors of variables: wff set class
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-ontowf1o 5197  cfv 5198  (class class class)co 5853  Fincfn 6718  0cc0 7774  1c1 7775   · cmul 7779  cle 7955   # cap 8500  cn 8878  cz 9212  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)
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