Theorem fprodfac 12121

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.

Step	Hyp	Ref	Expression
1		fveq2 5626	. . 3 ⊢ (𝑤 = 0 → (!‘𝑤) = (!‘0))
2		oveq2 6008	. . . 4 ⊢ (𝑤 = 0 → (1...𝑤) = (1...0))
3	2	prodeq1d 12070	. . 3 ⊢ (𝑤 = 0 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...0)𝑘)
4	1, 3	eqeq12d 2244	. 2 ⊢ (𝑤 = 0 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘0) = ∏𝑘 ∈ (1...0)𝑘))
5		fveq2 5626	. . 3 ⊢ (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6		oveq2 6008	. . . 4 ⊢ (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
7	6	prodeq1d 12070	. . 3 ⊢ (𝑤 = 𝑥 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝑥)𝑘)
8	5, 7	eqeq12d 2244	. 2 ⊢ (𝑤 = 𝑥 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘))
9		fveq2 5626	. . 3 ⊢ (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10		oveq2 6008	. . . 4 ⊢ (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
11	10	prodeq1d 12070	. . 3 ⊢ (𝑤 = (𝑥 + 1) → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
12	9, 11	eqeq12d 2244	. 2 ⊢ (𝑤 = (𝑥 + 1) → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
13		fveq2 5626	. . 3 ⊢ (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14		oveq2 6008	. . . 4 ⊢ (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
15	14	prodeq1d 12070	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝐴)𝑘)
16	13, 15	eqeq12d 2244	. 2 ⊢ (𝑤 = 𝐴 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘))
17		prod0 12091	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1
18		fz10 10238	. . . 4 ⊢ (1...0) = ∅
19	18	prodeq1i 12067	. . 3 ⊢ ∏𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20		fac0 10945	. . 3 ⊢ (!‘0) = 1
21	17, 19, 20	3eqtr4ri 2261	. 2 ⊢ (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22		elnn0 9367	. . 3 ⊢ (𝑥 ∈ ℕ0 ↔ (𝑥 ∈ ℕ ∨ 𝑥 = 0))
23		simpr 110	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘)
24	23	oveq1d 6015	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25		nnnn0 9372	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26		facp1 10947	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28		elnnuz 9755	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpi 120	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ≥‘1))
30		elfzelz 10217	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
31	30	zcnd 9566	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℂ)
32	31	adantl 277	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑥 + 1))) → 𝑘 ∈ ℂ)
33		id 19	. . . . . . . . 9 ⊢ (𝑘 = (𝑥 + 1) → 𝑘 = (𝑥 + 1))
34	29, 32, 33	fprodp1 12106	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
35	27, 34	eqeq12d 2244	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ((!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 ↔ ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1))))
36	35	adantr 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))))
37	24, 36	mpbird 167	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
38	37	ex 115	. . . 4 ⊢ (𝑥 ∈ ℕ → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
39		1zzd 9469	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℤ)
40		1cnd 8158	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℂ)
41		id 19	. . . . . . . 8 ⊢ (𝑘 = 1 → 𝑘 = 1)
42	41	fprod1 12100	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → ∏𝑘 ∈ (1...1)𝑘 = 1)
43	39, 40, 42	syl2anc 411	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...1)𝑘 = 1)
44		oveq1 6007	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45		0p1e1 9220	. . . . . . . . 9 ⊢ (0 + 1) = 1
46	44, 45	eqtrdi 2278	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
47	46	oveq2d 6016	. . . . . . 7 ⊢ (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
48	47	prodeq1d 12070	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49		fv0p1e1 9221	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50		fac1 10946	. . . . . . 7 ⊢ (!‘1) = 1
51	49, 50	eqtrdi 2278	. . . . . 6 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
52	43, 48, 51	3eqtr4rd 2273	. . . . 5 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
53	52	a1d 22	. . . 4 ⊢ (𝑥 = 0 → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
54	38, 53	jaoi 721	. . 3 ⊢ ((𝑥 ∈ ℕ ∨ 𝑥 = 0) → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
55	22, 54	sylbi 121	. 2 ⊢ (𝑥 ∈ ℕ0 → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
56	4, 8, 12, 16, 21, 55	nn0ind 9557	1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∅c0 3491  ‘cfv 5317  (class class class)co 6000  ℂcc 7993  0cc0 7995  1c1 7996   + caddc 7998   · cmul 8000  ℕcn 9106  ℕ₀cn0 9365  ℤcz 9442  ℤ_≥cuz 9718  ...cfz 10200  !cfa 10942  ∏cprod 12056

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-ihash 10993  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-proddc 12057

This theorem is referenced by:  gausslemma2dlem1  15734  gausslemma2dlem6  15740