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Theorem fprodfac 11758
Description: Factorial using product notation. (Contributed by Scott Fenton, 15-Dec-2017.)
Assertion
Ref Expression
fprodfac (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)
Distinct variable group:   𝐴,𝑘

Proof of Theorem fprodfac
Dummy variables 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5554 . . 3 (𝑤 = 0 → (!‘𝑤) = (!‘0))
2 oveq2 5926 . . . 4 (𝑤 = 0 → (1...𝑤) = (1...0))
32prodeq1d 11707 . . 3 (𝑤 = 0 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...0)𝑘)
41, 3eqeq12d 2208 . 2 (𝑤 = 0 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘0) = ∏𝑘 ∈ (1...0)𝑘))
5 fveq2 5554 . . 3 (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6 oveq2 5926 . . . 4 (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
76prodeq1d 11707 . . 3 (𝑤 = 𝑥 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝑥)𝑘)
85, 7eqeq12d 2208 . 2 (𝑤 = 𝑥 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘))
9 fveq2 5554 . . 3 (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10 oveq2 5926 . . . 4 (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
1110prodeq1d 11707 . . 3 (𝑤 = (𝑥 + 1) → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
129, 11eqeq12d 2208 . 2 (𝑤 = (𝑥 + 1) → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
13 fveq2 5554 . . 3 (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14 oveq2 5926 . . . 4 (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
1514prodeq1d 11707 . . 3 (𝑤 = 𝐴 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝐴)𝑘)
1613, 15eqeq12d 2208 . 2 (𝑤 = 𝐴 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘))
17 prod0 11728 . . 3 𝑘 ∈ ∅ 𝑘 = 1
18 fz10 10112 . . . 4 (1...0) = ∅
1918prodeq1i 11704 . . 3 𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20 fac0 10799 . . 3 (!‘0) = 1
2117, 19, 203eqtr4ri 2225 . 2 (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22 elnn0 9242 . . 3 (𝑥 ∈ ℕ0 ↔ (𝑥 ∈ ℕ ∨ 𝑥 = 0))
23 simpr 110 . . . . . . 7 ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘)
2423oveq1d 5933 . . . . . 6 ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25 nnnn0 9247 . . . . . . . . 9 (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26 facp1 10801 . . . . . . . . 9 (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
2725, 26syl 14 . . . . . . . 8 (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28 elnnuz 9629 . . . . . . . . . 10 (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ‘1))
2928biimpi 120 . . . . . . . . 9 (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ‘1))
30 elfzelz 10091 . . . . . . . . . . 11 (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
3130zcnd 9440 . . . . . . . . . 10 (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℂ)
3231adantl 277 . . . . . . . . 9 ((𝑥 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑥 + 1))) → 𝑘 ∈ ℂ)
33 id 19 . . . . . . . . 9 (𝑘 = (𝑥 + 1) → 𝑘 = (𝑥 + 1))
3429, 32, 33fprodp1 11743 . . . . . . . 8 (𝑥 ∈ ℕ → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
3527, 34eqeq12d 2208 . . . . . . 7 (𝑥 ∈ ℕ → ((!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 ↔ ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1))))
3635adantr 276 . . . . . 6 ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 ↔ ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1))))
3724, 36mpbird 167 . . . . 5 ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
3837ex 115 . . . 4 (𝑥 ∈ ℕ → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
39 1zzd 9344 . . . . . . 7 (𝑥 = 0 → 1 ∈ ℤ)
40 1cnd 8035 . . . . . . 7 (𝑥 = 0 → 1 ∈ ℂ)
41 id 19 . . . . . . . 8 (𝑘 = 1 → 𝑘 = 1)
4241fprod1 11737 . . . . . . 7 ((1 ∈ ℤ ∧ 1 ∈ ℂ) → ∏𝑘 ∈ (1...1)𝑘 = 1)
4339, 40, 42syl2anc 411 . . . . . 6 (𝑥 = 0 → ∏𝑘 ∈ (1...1)𝑘 = 1)
44 oveq1 5925 . . . . . . . . 9 (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45 0p1e1 9096 . . . . . . . . 9 (0 + 1) = 1
4644, 45eqtrdi 2242 . . . . . . . 8 (𝑥 = 0 → (𝑥 + 1) = 1)
4746oveq2d 5934 . . . . . . 7 (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
4847prodeq1d 11707 . . . . . 6 (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49 fv0p1e1 9097 . . . . . . 7 (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50 fac1 10800 . . . . . . 7 (!‘1) = 1
5149, 50eqtrdi 2242 . . . . . 6 (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
5243, 48, 513eqtr4rd 2237 . . . . 5 (𝑥 = 0 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
5352a1d 22 . . . 4 (𝑥 = 0 → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
5438, 53jaoi 717 . . 3 ((𝑥 ∈ ℕ ∨ 𝑥 = 0) → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
5522, 54sylbi 121 . 2 (𝑥 ∈ ℕ0 → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
564, 8, 12, 16, 21, 55nn0ind 9431 1 (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wcel 2164  c0 3446  cfv 5254  (class class class)co 5918  cc 7870  0cc0 7872  1c1 7873   + caddc 7875   · cmul 7877  cn 8982  0cn0 9240  cz 9317  cuz 9592  ...cfz 10074  !cfa 10796  cprod 11693
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990  ax-arch 7991  ax-caucvg 7992
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 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-isom 5263  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-frec 6444  df-1o 6469  df-oadd 6473  df-er 6587  df-en 6795  df-dom 6796  df-fin 6797  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-uz 9593  df-q 9685  df-rp 9720  df-fz 10075  df-fzo 10209  df-seqfrec 10519  df-exp 10610  df-fac 10797  df-ihash 10847  df-cj 10986  df-re 10987  df-im 10988  df-rsqrt 11142  df-abs 11143  df-clim 11422  df-proddc 11694
This theorem is referenced by:  gausslemma2dlem1  15177  gausslemma2dlem6  15183
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