Theorem 2expltfac 13241

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 6093	. . . 4 |- ( x = 4 -> ( 2 ^ x ) = ( 2 ^ 4 ) )
2		2exp4 13233	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
3	1, 2	eqtrdi 2287	. . 3 |- ( x = 4 -> ( 2 ^ x ) = ; 1 6 )
4		fveq2 5695	. . . 4 |- ( x = 4 -> ( ! ` x ) = ( ! ` 4 ) )
5		fac4 11186	. . . 4 |- ( ! ` 4 ) = ; 2 4
6	4, 5	eqtrdi 2287	. . 3 |- ( x = 4 -> ( ! ` x ) = ; 2 4 )
7	3, 6	breq12d 4143	. 2 |- ( x = 4 -> ( ( 2 ^ x ) < ( ! ` x ) <-> ; 1 6 < ; 2 4 ) )
8		oveq2 6093	. . 3 |- ( x = n -> ( 2 ^ x ) = ( 2 ^ n ) )
9		fveq2 5695	. . 3 |- ( x = n -> ( ! ` x ) = ( ! ` n ) )
10	8, 9	breq12d 4143	. 2 |- ( x = n -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ n ) < ( ! ` n ) ) )
11		oveq2 6093	. . 3 |- ( x = ( n + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( n + 1 ) ) )
12		fveq2 5695	. . 3 |- ( x = ( n + 1 ) -> ( ! ` x ) = ( ! ` ( n + 1 ) ) )
13	11, 12	breq12d 4143	. 2 |- ( x = ( n + 1 ) -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
14		oveq2 6093	. . 3 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
15		fveq2 5695	. . 3 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
16	14, 15	breq12d 4143	. 2 |- ( x = N -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ N ) < ( ! ` N ) ) )
17		1nn0 9584	. . 3 |- 1 e. NN0
18		2nn0 9585	. . 3 |- 2 e. NN0
19		6nn0 9589	. . 3 |- 6 e. NN0
20		4nn0 9587	. . 3 |- 4 e. NN0
21		6lt10 9920	. . 3 |- 6 < ; 1 0
22		1lt2 9479	. . 3 |- 1 < 2
23	17, 18, 19, 20, 21, 22	decltc 9815	. 2 |- ; 1 6 < ; 2 4
24		2nn 9471	. . . . . . . . 9 |- 2 e. NN
25	24	a1i 9	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. NN )
26		4nn 9473	. . . . . . . . . 10 |- 4 e. NN
27		simpl 109	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. ( ZZ>= ` 4 ) )
28		eluznn 10010	. . . . . . . . . 10 |- ( ( 4 e. NN /\ n e. ( ZZ>= ` 4 ) ) -> n e. NN )
29	26, 27, 28	sylancr 418	. . . . . . . . 9 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN )
30	29	nnnn0d 9625	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN0 )
31	25, 30	nnexpcld 11147	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. NN )
32	31	nnred 9320	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. RR )
33		2re 9377	. . . . . . 7 |- 2 e. RR
34	33	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR )
35	32, 34	remulcld 8357	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) e. RR )
36	30	faccld 11189	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN )
37	36	nnred 9320	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. RR )
38	37, 34	remulcld 8357	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) e. RR )
39	29	nnred 9320	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. RR )
40		1red 8342	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 e. RR )
41	39, 40	readdcld 8356	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( n + 1 ) e. RR )
42	37, 41	remulcld 8357	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. ( n + 1 ) ) e. RR )
43		2rp 10070	. . . . . . 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 10164	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. 2 ) )
47	36	nnnn0d 9625	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN0 )
48	47	nn0ge0d 9628	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 0 <_ ( ! ` n ) )
49		df-2 9366	. . . . . . 7 |- 2 = ( 1 + 1 )
50	29	nnge1d 9350	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 <_ n )
51	40, 39, 40, 50	leadd1dd 8889	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 1 + 1 ) <_ ( n + 1 ) )
52	49, 51	eqbrtrid 4165	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 <_ ( n + 1 ) )
53	34, 41, 37, 48, 52	lemul2ad 9273	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) <_ ( ( ! ` n ) x. ( n + 1 ) ) )
54	35, 38, 42, 46, 53	ltletrd 8753	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. ( n + 1 ) ) )
55		2cnd 9380	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. CC )
56	55, 30	expp1d 11126	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) = ( ( 2 ^ n ) x. 2 ) )
57		facp1 11183	. . . . 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 4162	. . 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 10002	1 |- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   RRcr 8179   1c1 8181   + caddc 8183   x. cmul 8185   < clt 8361   <_ cle 8362   NNcn 9307   2c2 9358   4c4 9360   6c6 9362   NN0cn0 9568  ;cdc 9782   ZZ>=cuz 9931   RR+crp 10065   ^cexp 10989   !cfa 11178

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-uz 9932  df-rp 10066  df-seqfrec 10899  df-exp 10990  df-fac 11179

This theorem is used by: (None)