Theorem fprodfac 11607

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 5511	. . 3 ⊢ (𝑤 = 0 → (!‘𝑤) = (!‘0))
2		oveq2 5877	. . . 4 ⊢ (𝑤 = 0 → (1...𝑤) = (1...0))
3	2	prodeq1d 11556	. . 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 5511	. . 3 ⊢ (𝑤 = 𝑥 → (!‘𝑤) = (!‘𝑥))
6		oveq2 5877	. . . 4 ⊢ (𝑤 = 𝑥 → (1...𝑤) = (1...𝑥))
7	6	prodeq1d 11556	. . 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 5511	. . 3 ⊢ (𝑤 = (𝑥 + 1) → (!‘𝑤) = (!‘(𝑥 + 1)))
10		oveq2 5877	. . . 4 ⊢ (𝑤 = (𝑥 + 1) → (1...𝑤) = (1...(𝑥 + 1)))
11	10	prodeq1d 11556	. . 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 5511	. . 3 ⊢ (𝑤 = 𝐴 → (!‘𝑤) = (!‘𝐴))
14		oveq2 5877	. . . 4 ⊢ (𝑤 = 𝐴 → (1...𝑤) = (1...𝐴))
15	14	prodeq1d 11556	. . 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 11577	. . 3 ⊢ ∏𝑘 ∈ ∅ 𝑘 = 1
18		fz10 10032	. . . 4 ⊢ (1...0) = ∅
19	18	prodeq1i 11553	. . 3 ⊢ ∏𝑘 ∈ (1...0)𝑘 = ∏𝑘 ∈ ∅ 𝑘
20		fac0 10692	. . 3 ⊢ (!‘0) = 1
21	17, 19, 20	3eqtr4ri 2209	. 2 ⊢ (!‘0) = ∏𝑘 ∈ (1...0)𝑘
22		elnn0 9167	. . 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 5884	. . . . . 6 ⊢ ((𝑥 ∈ ℕ ∧ (!‘𝑥) = ∏𝑘 ∈ (1...𝑥)𝑘) → ((!‘𝑥) · (𝑥 + 1)) = (∏𝑘 ∈ (1...𝑥)𝑘 · (𝑥 + 1)))
25		nnnn0 9172	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℕ0)
26		facp1 10694	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ0 → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
27	25, 26	syl 14	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ → (!‘(𝑥 + 1)) = ((!‘𝑥) · (𝑥 + 1)))
28		elnnuz 9553	. . . . . . . . . 10 ⊢ (𝑥 ∈ ℕ ↔ 𝑥 ∈ (ℤ≥‘1))
29	28	biimpi 120	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ (ℤ≥‘1))
30		elfzelz 10011	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (1...(𝑥 + 1)) → 𝑘 ∈ ℤ)
31	30	zcnd 9365	. . . . . . . . . 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 11592	. . . . . . . 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 9269	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℤ)
40		1cnd 7964	. . . . . . 7 ⊢ (𝑥 = 0 → 1 ∈ ℂ)
41		id 19	. . . . . . . 8 ⊢ (𝑘 = 1 → 𝑘 = 1)
42	41	fprod1 11586	. . . . . . 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 5876	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1))
45		0p1e1 9022	. . . . . . . . 9 ⊢ (0 + 1) = 1
46	44, 45	eqtrdi 2226	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 + 1) = 1)
47	46	oveq2d 5885	. . . . . . 7 ⊢ (𝑥 = 0 → (1...(𝑥 + 1)) = (1...1))
48	47	prodeq1d 11556	. . . . . 6 ⊢ (𝑥 = 0 → ∏𝑘 ∈ (1...(𝑥 + 1))𝑘 = ∏𝑘 ∈ (1...1)𝑘)
49		fv0p1e1 9023	. . . . . . 7 ⊢ (𝑥 = 0 → (!‘(𝑥 + 1)) = (!‘1))
50		fac1 10693	. . . . . . 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 9356	1 ⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ (1...𝐴)𝑘)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∅c0 3422  ‘cfv 5212  (class class class)co 5869  ℂcc 7800  0cc0 7802  1c1 7803   + caddc 7805   · cmul 7807  ℕcn 8908  ℕ₀cn0 9165  ℤcz 9242  ℤ_≥cuz 9517  ...cfz 9995  !cfa 10689  ∏cprod 11542

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-fac 10690  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543

This theorem is referenced by: (None)