Theorem fprodfac 11556

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 5486	. . 3 ⊢ (𝑤 = 0 → (!‘𝑤) = (!‘0))
2		oveq2 5850	. . . 4 ⊢ (𝑤 = 0 → (1...𝑤) = (1...0))
3	2	prodeq1d 11505	. . 3 ⊢ (𝑤 = 0 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...0)𝑘)
4	1, 3	eqeq12d 2180	. 2 ⊢ (𝑤 = 0 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘0) = ∏𝑘 ∈ (1...0)𝑘))
5		fveq2 5486	. . 3 ⊢ (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6		oveq2 5850	. . . 4 ⊢ (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
7	6	prodeq1d 11505	. . 3 ⊢ (𝑤 = 𝑥 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝑥)𝑘)
8	5, 7	eqeq12d 2180	. 2 ⊢ (𝑤 = 𝑥 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘))
9		fveq2 5486	. . 3 ⊢ (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10		oveq2 5850	. . . 4 ⊢ (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
11	10	prodeq1d 11505	. . 3 ⊢ (𝑤 = (𝑥 + 1) → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
12	9, 11	eqeq12d 2180	. 2 ⊢ (𝑤 = (𝑥 + 1) → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
13		fveq2 5486	. . 3 ⊢ (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14		oveq2 5850	. . . 4 ⊢ (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
15	14	prodeq1d 11505	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝐴)𝑘)
16	13, 15	eqeq12d 2180	. 2 ⊢ (𝑤 = 𝐴 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘))
17		prod0 11526	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1
18		fz10 9981	. . . 4 ⊢ (1...0) = ∅
19	18	prodeq1i 11502	. . 3 ⊢ ∏𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20		fac0 10641	. . 3 ⊢ (!‘0) = 1
21	17, 19, 20	3eqtr4ri 2197	. 2 ⊢ (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22		elnn0 9116	. . 3 ⊢ (𝑥 ∈ ℕ0 ↔ (𝑥 ∈ ℕ ∨ 𝑥 = 0))
23		simpr 109	. . . . . . 7 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘)
24	23	oveq1d 5857	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25		nnnn0 9121	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26		facp1 10643	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28		elnnuz 9502	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpi 119	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ≥‘1))
30		elfzelz 9960	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
31	30	zcnd 9314	. . . . . . . . . 10 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℂ)
32	31	adantl 275	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℕ ∧ 𝑘 ∈ (1...(𝑥 + 1))) → 𝑘 ∈ ℂ)
33		id 19	. . . . . . . . 9 ⊢ (𝑘 = (𝑥 + 1) → 𝑘 = (𝑥 + 1))
34	29, 32, 33	fprodp1 11541	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
35	27, 34	eqeq12d 2180	. . . . . . 7 ⊢ (𝑥 ∈ ℕ → ((!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 ↔ ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1))))
36	35	adantr 274	. . . . . 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 166	. . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
38	37	ex 114	. . . 4 ⊢ (𝑥 ∈ ℕ → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
39		1zzd 9218	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℤ)
40		1cnd 7915	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℂ)
41		id 19	. . . . . . . 8 ⊢ (𝑘 = 1 → 𝑘 = 1)
42	41	fprod1 11535	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → ∏𝑘 ∈ (1...1)𝑘 = 1)
43	39, 40, 42	syl2anc 409	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...1)𝑘 = 1)
44		oveq1 5849	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45		0p1e1 8971	. . . . . . . . 9 ⊢ (0 + 1) = 1
46	44, 45	eqtrdi 2215	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
47	46	oveq2d 5858	. . . . . . 7 ⊢ (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
48	47	prodeq1d 11505	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49		fv0p1e1 8972	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50		fac1 10642	. . . . . . 7 ⊢ (!‘1) = 1
51	49, 50	eqtrdi 2215	. . . . . 6 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
52	43, 48, 51	3eqtr4rd 2209	. . . . 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 706	. . 3 ⊢ ((𝑥 ∈ ℕ ∨ 𝑥 = 0) → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
55	22, 54	sylbi 120	. 2 ⊢ (𝑥 ∈ ℕ0 → ((!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘 → (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
56	4, 8, 12, 16, 21, 55	nn0ind 9305	1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1343   ∈ wcel 2136  ∅c0 3409  ‘cfv 5188  (class class class)co 5842  ℂcc 7751  0cc0 7753  1c1 7754   + caddc 7756   · cmul 7758  ℕcn 8857  ℕ₀cn0 9114  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  !cfa 10638  ∏cprod 11491

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-proddc 11492

This theorem is referenced by: (None)