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Theorem prod1dc 11578
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 2177 . . 3 (ℤ𝑀) = (ℤ𝑀)
2 simp1 997 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 𝑀 ∈ ℤ)
3 1ap0 8537 . . . 4 1 # 0
43a1i 9 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 1 # 0)
51prodfclim1 11536 . . . 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 999 . . . 4 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴)
8 eleq1w 2238 . . . . . 6 (𝑗 = 𝑎 → (𝑗𝐴𝑎𝐴))
98dcbid 838 . . . . 5 (𝑗 = 𝑎 → (DECID 𝑗𝐴DECID 𝑎𝐴))
109cbvralv 2703 . . . 4 (∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴 ↔ ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
117, 10sylib 122 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
12 simp2 998 . . 3 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → 𝐴 ⊆ (ℤ𝑀))
13 1ex 7943 . . . . . 6 1 ∈ V
1413fvconst2 5728 . . . . 5 (𝑘 ∈ (ℤ𝑀) → (((ℤ𝑀) × {1})‘𝑘) = 1)
1514adantl 277 . . . 4 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → (((ℤ𝑀) × {1})‘𝑘) = 1)
16 eleq1w 2238 . . . . . . 7 (𝑎 = 𝑘 → (𝑎𝐴𝑘𝐴))
1716dcbid 838 . . . . . 6 (𝑎 = 𝑘 → (DECID 𝑎𝐴DECID 𝑘𝐴))
1811adantr 276 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → ∀𝑎 ∈ (ℤ𝑀)DECID 𝑎𝐴)
19 simpr 110 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → 𝑘 ∈ (ℤ𝑀))
2017, 18, 19rspcdva 2846 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → DECID 𝑘𝐴)
21 ifiddc 3567 . . . . 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 2213 . . 3 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘 ∈ (ℤ𝑀)) → (((ℤ𝑀) × {1})‘𝑘) = if(𝑘𝐴, 1, 1))
24 1cnd 7964 . . 3 (((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) ∧ 𝑘𝐴) → 1 ∈ ℂ)
251, 2, 4, 6, 11, 12, 23, 24zprodap0 11573 . 2 ((𝑀 ∈ ℤ ∧ 𝐴 ⊆ (ℤ𝑀) ∧ ∀𝑗 ∈ (ℤ𝑀)DECID 𝑗𝐴) → ∏𝑘𝐴 1 = 1)
26 fz1f1o 11367 . . 3 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)))
27 prodeq1 11545 . . . . 5 (𝐴 = ∅ → ∏𝑘𝐴 1 = ∏𝑘 ∈ ∅ 1)
28 prod0 11577 . . . . 5 𝑘 ∈ ∅ 1 = 1
2927, 28eqtrdi 2226 . . . 4 (𝐴 = ∅ → ∏𝑘𝐴 1 = 1)
30 eqidd 2178 . . . . . . . . . 10 (𝑘 = (𝑓𝑗) → 1 = 1)
31 simpl 109 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (♯‘𝐴) ∈ ℕ)
32 simpr 110 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴)
33 1cnd 7964 . . . . . . . . . 10 ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ 𝑘𝐴) → 1 ∈ ℂ)
34 elfznn 10040 . . . . . . . . . . . 12 (𝑗 ∈ (1...(♯‘𝐴)) → 𝑗 ∈ ℕ)
3513fvconst2 5728 . . . . . . . . . . . 12 (𝑗 ∈ ℕ → ((ℕ × {1})‘𝑗) = 1)
3634, 35syl 14 . . . . . . . . . . 11 (𝑗 ∈ (1...(♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
3736adantl 277 . . . . . . . . . 10 ((((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) ∧ 𝑗 ∈ (1...(♯‘𝐴))) → ((ℕ × {1})‘𝑗) = 1)
3830, 31, 32, 33, 37fprodseq 11575 . . . . . . . . 9 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = (seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)))‘(♯‘𝐴)))
39 simpr 110 . . . . . . . . . . . . . . . . 17 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → 𝑗 ≤ (♯‘𝐴))
4039iftrued 3541 . . . . . . . . . . . . . . . 16 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = ((ℕ × {1})‘𝑗))
4135ad2antlr 489 . . . . . . . . . . . . . . . 16 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ 𝑗 ≤ (♯‘𝐴)) → ((ℕ × {1})‘𝑗) = 1)
4240, 41eqtrd 2210 . . . . . . . . . . . . . . 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))
4443iffalsed 3544 . . . . . . . . . . . . . . 15 ((((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑗 ≤ (♯‘𝐴)) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
45 nnz 9261 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ ℕ → 𝑗 ∈ ℤ)
46 nnz 9261 . . . . . . . . . . . . . . . . 17 ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℤ)
47 zdcle 9318 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → DECID 𝑗 ≤ (♯‘𝐴))
4845, 46, 47syl2anr 290 . . . . . . . . . . . . . . . 16 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → DECID 𝑗 ≤ (♯‘𝐴))
49 exmiddc 836 . . . . . . . . . . . . . . . 16 (DECID 𝑗 ≤ (♯‘𝐴) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
5048, 49syl 14 . . . . . . . . . . . . . . 15 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → (𝑗 ≤ (♯‘𝐴) ∨ ¬ 𝑗 ≤ (♯‘𝐴)))
5142, 44, 50mpjaodan 798 . . . . . . . . . . . . . 14 (((♯‘𝐴) ∈ ℕ ∧ 𝑗 ∈ ℕ) → if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1) = 1)
5251mpteq2dva 4090 . . . . . . . . . . . . 13 ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (𝑗 ∈ ℕ ↦ 1))
53 fconstmpt 4670 . . . . . . . . . . . . 13 (ℕ × {1}) = (𝑗 ∈ ℕ ↦ 1)
5452, 53eqtr4di 2228 . . . . . . . . . . . 12 ((♯‘𝐴) ∈ ℕ → (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1)) = (ℕ × {1}))
5554seqeq3d 10439 . . . . . . . . . . 11 ((♯‘𝐴) ∈ ℕ → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
5655adantr 276 . . . . . . . . . 10 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → seq1( · , (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ((ℕ × {1})‘𝑗), 1))) = seq1( · , (ℕ × {1})))
5756fveq1d 5513 . . . . . . . . 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 2210 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = (seq1( · , (ℕ × {1}))‘(♯‘𝐴)))
59 nnuz 9552 . . . . . . . . . 10 ℕ = (ℤ‘1)
6059prodf1 11534 . . . . . . . . 9 ((♯‘𝐴) ∈ ℕ → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
6160adantr 276 . . . . . . . 8 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → (seq1( · , (ℕ × {1}))‘(♯‘𝐴)) = 1)
6258, 61eqtrd 2210 . . . . . . 7 (((♯‘𝐴) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = 1)
6362ex 115 . . . . . 6 ((♯‘𝐴) ∈ ℕ → (𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 1 = 1))
6463exlimdv 1819 . . . . 5 ((♯‘𝐴) ∈ ℕ → (∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴 → ∏𝑘𝐴 1 = 1))
6564imp 124 . . . 4 (((♯‘𝐴) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘𝐴))–1-1-onto𝐴) → ∏𝑘𝐴 1 = 1)
6629, 65jaoi 716 . . 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 716 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 104  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  wss 3129  c0 3422  ifcif 3534  {csn 3591   class class class wbr 4000  cmpt 4061   × cxp 4621  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  0cc0 7802  1c1 7803   · cmul 7807  cle 7983   # cap 8528  cn 8908  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  chash 10739  cli 11270  cprod 11542
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543
This theorem is referenced by: (None)
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