Theorem fprodfac 12360

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 5690	. . 3 ⊢ (𝑤 = 0 → (!‘𝑤) = (!‘0))
2		oveq2 6083	. . . 4 ⊢ (𝑤 = 0 → (1...𝑤) = (1...0))
3	2	prodeq1d 12309	. . 3 ⊢ (𝑤 = 0 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...0)𝑘)
4	1, 3	eqeq12d 2253	. 2 ⊢ (𝑤 = 0 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘0) = ∏𝑘 ∈ (1...0)𝑘))
5		fveq2 5690	. . 3 ⊢ (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6		oveq2 6083	. . . 4 ⊢ (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
7	6	prodeq1d 12309	. . 3 ⊢ (𝑤 = 𝑥 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝑥)𝑘)
8	5, 7	eqeq12d 2253	. 2 ⊢ (𝑤 = 𝑥 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘))
9		fveq2 5690	. . 3 ⊢ (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10		oveq2 6083	. . . 4 ⊢ (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
11	10	prodeq1d 12309	. . 3 ⊢ (𝑤 = (𝑥 + 1) → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
12	9, 11	eqeq12d 2253	. 2 ⊢ (𝑤 = (𝑥 + 1) → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
13		fveq2 5690	. . 3 ⊢ (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14		oveq2 6083	. . . 4 ⊢ (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
15	14	prodeq1d 12309	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝐴)𝑘)
16	13, 15	eqeq12d 2253	. 2 ⊢ (𝑤 = 𝐴 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘))
17		prod0 12330	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1
18		fz10 10429	. . . 4 ⊢ (1...0) = ∅
19	18	prodeq1i 12306	. . 3 ⊢ ∏𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20		fac0 11144	. . 3 ⊢ (!‘0) = 1
21	17, 19, 20	3eqtr4ri 2270	. 2 ⊢ (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22		elnn0 9544	. . 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 6090	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25		nnnn0 9549	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26		facp1 11146	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28		elnnuz 9938	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpi 120	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ≥‘1))
30		elfzelz 10407	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
31	30	zcnd 9748	. . . . . . . . . 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 12345	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
35	27, 34	eqeq12d 2253	. . . . . . 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 9650	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℤ)
40		1cnd 8332	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℂ)
41		id 19	. . . . . . . 8 ⊢ (𝑘 = 1 → 𝑘 = 1)
42	41	fprod1 12339	. . . . . . 7 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℂ) → ∏𝑘 ∈ (1...1)𝑘 = 1)
43	39, 40, 42	syl2anc 415	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...1)𝑘 = 1)
44		oveq1 6082	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45		0p1e1 9397	. . . . . . . . 9 ⊢ (0 + 1) = 1
46	44, 45	eqtrdi 2287	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
47	46	oveq2d 6091	. . . . . . 7 ⊢ (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
48	47	prodeq1d 12309	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49		fv0p1e1 9398	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50		fac1 11145	. . . . . . 7 ⊢ (!‘1) = 1
51	49, 50	eqtrdi 2287	. . . . . 6 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
52	43, 48, 51	3eqtr4rd 2282	. . . . 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 728	. . 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 9739	1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∅c0 3520  ‘cfv 5372  (class class class)co 6075  ℂcc 8167  0cc0 8169  1c1 8170   + caddc 8172   · cmul 8174  ℕcn 9283  ℕ₀cn0 9542  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  !cfa 11141  ∏cprod 12295

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-ihash 11193  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-proddc 12296

This theorem is referenced by:  gausslemma2dlem1  16094  gausslemma2dlem6  16100