Theorem 2expltfac 12832

Description: The factorial grows faster than two to the power N. (Contributed by Mario Carneiro, 15-Sep-2016.)

Assertion

Ref	Expression
2expltfac	|- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

Proof of Theorem 2expltfac

Dummy variables x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5964	. . . 4 |- ( x = 4 -> ( 2 ^ x ) = ( 2 ^ 4 ) )
2		2exp4 12824	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
3	1, 2	eqtrdi 2255	. . 3 |- ( x = 4 -> ( 2 ^ x ) = ; 1 6 )
4		fveq2 5588	. . . 4 |- ( x = 4 -> ( ! ` x ) = ( ! ` 4 ) )
5		fac4 10895	. . . 4 |- ( ! ` 4 ) = ; 2 4
6	4, 5	eqtrdi 2255	. . 3 |- ( x = 4 -> ( ! ` x ) = ; 2 4 )
7	3, 6	breq12d 4063	. 2 |- ( x = 4 -> ( ( 2 ^ x ) < ( ! ` x ) <-> ; 1 6 < ; 2 4 ) )
8		oveq2 5964	. . 3 |- ( x = n -> ( 2 ^ x ) = ( 2 ^ n ) )
9		fveq2 5588	. . 3 |- ( x = n -> ( ! ` x ) = ( ! ` n ) )
10	8, 9	breq12d 4063	. 2 |- ( x = n -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ n ) < ( ! ` n ) ) )
11		oveq2 5964	. . 3 |- ( x = ( n + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( n + 1 ) ) )
12		fveq2 5588	. . 3 |- ( x = ( n + 1 ) -> ( ! ` x ) = ( ! ` ( n + 1 ) ) )
13	11, 12	breq12d 4063	. 2 |- ( x = ( n + 1 ) -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
14		oveq2 5964	. . 3 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
15		fveq2 5588	. . 3 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
16	14, 15	breq12d 4063	. 2 |- ( x = N -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ N ) < ( ! ` N ) ) )
17		1nn0 9326	. . 3 |- 1 e. NN0
18		2nn0 9327	. . 3 |- 2 e. NN0
19		6nn0 9331	. . 3 |- 6 e. NN0
20		4nn0 9329	. . 3 |- 4 e. NN0
21		6lt10 9652	. . 3 |- 6 < ; 1 0
22		1lt2 9221	. . 3 |- 1 < 2
23	17, 18, 19, 20, 21, 22	decltc 9547	. 2 |- ; 1 6 < ; 2 4
24		2nn 9213	. . . . . . . . 9 |- 2 e. NN
25	24	a1i 9	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. NN )
26		4nn 9215	. . . . . . . . . 10 |- 4 e. NN
27		simpl 109	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. ( ZZ>= ` 4 ) )
28		eluznn 9736	. . . . . . . . . 10 |- ( ( 4 e. NN /\ n e. ( ZZ>= ` 4 ) ) -> n e. NN )
29	26, 27, 28	sylancr 414	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN )
30	29	nnnn0d 9363	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN0 )
31	25, 30	nnexpcld 10857	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. NN )
32	31	nnred 9064	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. RR )
33		2re 9121	. . . . . . 7 |- 2 e. RR
34	33	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR )
35	32, 34	remulcld 8118	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) e. RR )
36	30	faccld 10898	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN )
37	36	nnred 9064	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. RR )
38	37, 34	remulcld 8118	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) e. RR )
39	29	nnred 9064	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. RR )
40		1red 8102	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 e. RR )
41	39, 40	readdcld 8117	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( n + 1 ) e. RR )
42	37, 41	remulcld 8118	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. ( n + 1 ) ) e. RR )
43		2rp 9795	. . . . . . 7 |- 2 e. RR+
44	43	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR+ )
45		simpr 110	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) < ( ! ` n ) )
46	32, 37, 44, 45	ltmul1dd 9889	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. 2 ) )
47	36	nnnn0d 9363	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN0 )
48	47	nn0ge0d 9366	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 0 <_ ( ! ` n ) )
49		df-2 9110	. . . . . . 7 |- 2 = ( 1 + 1 )
50	29	nnge1d 9094	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 <_ n )
51	40, 39, 40, 50	leadd1dd 8647	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 1 + 1 ) <_ ( n + 1 ) )
52	49, 51	eqbrtrid 4085	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 <_ ( n + 1 ) )
53	34, 41, 37, 48, 52	lemul2ad 9028	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) <_ ( ( ! ` n ) x. ( n + 1 ) ) )
54	35, 38, 42, 46, 53	ltletrd 8511	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. ( n + 1 ) ) )
55		2cnd 9124	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. CC )
56	55, 30	expp1d 10836	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) = ( ( 2 ^ n ) x. 2 ) )
57		facp1 10892	. . . . 5 |- ( n e. NN0 -> ( ! ` ( n + 1 ) ) = ( ( ! ` n ) x. ( n + 1 ) ) )
58	30, 57	syl 14	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` ( n + 1 ) ) = ( ( ! ` n ) x. ( n + 1 ) ) )
59	54, 56, 58	3brtr4d 4082	. . 3 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) )
60	59	ex 115	. 2 |- ( n e. ( ZZ>= ` 4 ) -> ( ( 2 ^ n ) < ( ! ` n ) -> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
61	7, 10, 13, 16, 23, 60	uzind4i 9728	1 |- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   class class class wbr 4050   ` cfv 5279  (class class class)co 5956   RRcr 7939   1c1 7941   + caddc 7943   x. cmul 7945   < clt 8122   <_ cle 8123   NNcn 9051   2c2 9102   4c4 9104   6c6 9106   NN0cn0 9310  ;cdc 9519   ZZ>=cuz 9663   RR+crp 9790   ^cexp 10700   !cfa 10887

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643  ax-cnex 8031  ax-resscn 8032  ax-1cn 8033  ax-1re 8034  ax-icn 8035  ax-addcl 8036  ax-addrcl 8037  ax-mulcl 8038  ax-mulrcl 8039  ax-addcom 8040  ax-mulcom 8041  ax-addass 8042  ax-mulass 8043  ax-distr 8044  ax-i2m1 8045  ax-0lt1 8046  ax-1rid 8047  ax-0id 8048  ax-rnegex 8049  ax-precex 8050  ax-cnre 8051  ax-pre-ltirr 8052  ax-pre-ltwlin 8053  ax-pre-lttrn 8054  ax-pre-apti 8055  ax-pre-ltadd 8056  ax-pre-mulgt0 8057  ax-pre-mulext 8058

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-ilim 4423  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-riota 5911  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-frec 6489  df-pnf 8124  df-mnf 8125  df-xr 8126  df-ltxr 8127  df-le 8128  df-sub 8260  df-neg 8261  df-reap 8663  df-ap 8670  df-div 8761  df-inn 9052  df-2 9110  df-3 9111  df-4 9112  df-5 9113  df-6 9114  df-7 9115  df-8 9116  df-9 9117  df-n0 9311  df-z 9388  df-dec 9520  df-uz 9664  df-rp 9791  df-seqfrec 10610  df-exp 10701  df-fac 10888

This theorem is referenced by: (None)