Theorem fprodfac 11623

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 5516	. . 3 ⊢ (𝑤 = 0 → (!‘𝑤) = (!‘0))
2		oveq2 5883	. . . 4 ⊢ (𝑤 = 0 → (1...𝑤) = (1...0))
3	2	prodeq1d 11572	. . 3 ⊢ (𝑤 = 0 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...0)𝑘)
4	1, 3	eqeq12d 2192	. 2 ⊢ (𝑤 = 0 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘0) = ∏𝑘 ∈ (1...0)𝑘))
5		fveq2 5516	. . 3 ⊢ (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6		oveq2 5883	. . . 4 ⊢ (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
7	6	prodeq1d 11572	. . 3 ⊢ (𝑤 = 𝑥 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝑥)𝑘)
8	5, 7	eqeq12d 2192	. 2 ⊢ (𝑤 = 𝑥 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘))
9		fveq2 5516	. . 3 ⊢ (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10		oveq2 5883	. . . 4 ⊢ (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
11	10	prodeq1d 11572	. . 3 ⊢ (𝑤 = (𝑥 + 1) → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘)
12	9, 11	eqeq12d 2192	. 2 ⊢ (𝑤 = (𝑥 + 1) → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘(𝑥 + 1)) = ∏𝑘 ∈ (1...(𝑥 + 1))𝑘))
13		fveq2 5516	. . 3 ⊢ (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14		oveq2 5883	. . . 4 ⊢ (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
15	14	prodeq1d 11572	. . 3 ⊢ (𝑤 = 𝐴 → ∏𝑘 ∈ (1...𝑤)𝑘 = ∏𝑘 ∈ (1...𝐴)𝑘)
16	13, 15	eqeq12d 2192	. 2 ⊢ (𝑤 = 𝐴 → ((!‘𝑤) = ∏𝑘 ∈ (1...𝑤)𝑘 ↔ (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘))
17		prod0 11593	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1
18		fz10 10046	. . . 4 ⊢ (1...0) = ∅
19	18	prodeq1i 11569	. . 3 ⊢ ∏𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20		fac0 10708	. . 3 ⊢ (!‘0) = 1
21	17, 19, 20	3eqtr4ri 2209	. 2 ⊢ (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22		elnn0 9178	. . 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 5890	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25		nnnn0 9183	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26		facp1 10710	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28		elnnuz 9564	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpi 120	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ≥‘1))
30		elfzelz 10025	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
31	30	zcnd 9376	. . . . . . . . . 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 11608	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
35	27, 34	eqeq12d 2192	. . . . . . 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 9280	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℤ)
40		1cnd 7973	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℂ)
41		id 19	. . . . . . . 8 ⊢ (𝑘 = 1 → 𝑘 = 1)
42	41	fprod1 11602	. . . . . . 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 5882	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45		0p1e1 9033	. . . . . . . . 9 ⊢ (0 + 1) = 1
46	44, 45	eqtrdi 2226	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
47	46	oveq2d 5891	. . . . . . 7 ⊢ (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
48	47	prodeq1d 11572	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49		fv0p1e1 9034	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50		fac1 10709	. . . . . . 7 ⊢ (!‘1) = 1
51	49, 50	eqtrdi 2226	. . . . . 6 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = 1)
52	43, 48, 51	3eqtr4rd 2221	. . . . 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 716	. . 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 9367	1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∅c0 3423  ‘cfv 5217  (class class class)co 5875  ℂcc 7809  0cc0 7811  1c1 7812   + caddc 7814   · cmul 7816  ℕcn 8919  ℕ₀cn0 9176  ℤcz 9253  ℤ_≥cuz 9528  ...cfz 10008  !cfa 10705  ∏cprod 11558

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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929  ax-arch 7930  ax-caucvg 7931

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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-po 4297  df-iso 4298  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-isom 5226  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-irdg 6371  df-frec 6392  df-1o 6417  df-oadd 6421  df-er 6535  df-en 6741  df-dom 6742  df-fin 6743  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-2 8978  df-3 8979  df-4 8980  df-n0 9177  df-z 9254  df-uz 9529  df-q 9620  df-rp 9654  df-fz 10009  df-fzo 10143  df-seqfrec 10446  df-exp 10520  df-fac 10706  df-ihash 10756  df-cj 10851  df-re 10852  df-im 10853  df-rsqrt 11007  df-abs 11008  df-clim 11287  df-proddc 11559

This theorem is referenced by: (None)