Theorem 2expltfac 12948

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 6002	. . . 4 |- ( x = 4 -> ( 2 ^ x ) = ( 2 ^ 4 ) )
2		2exp4 12940	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
3	1, 2	eqtrdi 2278	. . 3 |- ( x = 4 -> ( 2 ^ x ) = ; 1 6 )
4		fveq2 5623	. . . 4 |- ( x = 4 -> ( ! ` x ) = ( ! ` 4 ) )
5		fac4 10942	. . . 4 |- ( ! ` 4 ) = ; 2 4
6	4, 5	eqtrdi 2278	. . 3 |- ( x = 4 -> ( ! ` x ) = ; 2 4 )
7	3, 6	breq12d 4095	. 2 |- ( x = 4 -> ( ( 2 ^ x ) < ( ! ` x ) <-> ; 1 6 < ; 2 4 ) )
8		oveq2 6002	. . 3 |- ( x = n -> ( 2 ^ x ) = ( 2 ^ n ) )
9		fveq2 5623	. . 3 |- ( x = n -> ( ! ` x ) = ( ! ` n ) )
10	8, 9	breq12d 4095	. 2 |- ( x = n -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ n ) < ( ! ` n ) ) )
11		oveq2 6002	. . 3 |- ( x = ( n + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( n + 1 ) ) )
12		fveq2 5623	. . 3 |- ( x = ( n + 1 ) -> ( ! ` x ) = ( ! ` ( n + 1 ) ) )
13	11, 12	breq12d 4095	. 2 |- ( x = ( n + 1 ) -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
14		oveq2 6002	. . 3 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
15		fveq2 5623	. . 3 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
16	14, 15	breq12d 4095	. 2 |- ( x = N -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ N ) < ( ! ` N ) ) )
17		1nn0 9373	. . 3 |- 1 e. NN0
18		2nn0 9374	. . 3 |- 2 e. NN0
19		6nn0 9378	. . 3 |- 6 e. NN0
20		4nn0 9376	. . 3 |- 4 e. NN0
21		6lt10 9699	. . 3 |- 6 < ; 1 0
22		1lt2 9268	. . 3 |- 1 < 2
23	17, 18, 19, 20, 21, 22	decltc 9594	. 2 |- ; 1 6 < ; 2 4
24		2nn 9260	. . . . . . . . 9 |- 2 e. NN
25	24	a1i 9	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. NN )
26		4nn 9262	. . . . . . . . . 10 |- 4 e. NN
27		simpl 109	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. ( ZZ>= ` 4 ) )
28		eluznn 9783	. . . . . . . . . 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 9410	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN0 )
31	25, 30	nnexpcld 10904	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. NN )
32	31	nnred 9111	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. RR )
33		2re 9168	. . . . . . 7 |- 2 e. RR
34	33	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR )
35	32, 34	remulcld 8165	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) e. RR )
36	30	faccld 10945	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN )
37	36	nnred 9111	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. RR )
38	37, 34	remulcld 8165	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) e. RR )
39	29	nnred 9111	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. RR )
40		1red 8149	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 e. RR )
41	39, 40	readdcld 8164	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( n + 1 ) e. RR )
42	37, 41	remulcld 8165	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. ( n + 1 ) ) e. RR )
43		2rp 9842	. . . . . . 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 9936	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. 2 ) )
47	36	nnnn0d 9410	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN0 )
48	47	nn0ge0d 9413	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 0 <_ ( ! ` n ) )
49		df-2 9157	. . . . . . 7 |- 2 = ( 1 + 1 )
50	29	nnge1d 9141	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 <_ n )
51	40, 39, 40, 50	leadd1dd 8694	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 1 + 1 ) <_ ( n + 1 ) )
52	49, 51	eqbrtrid 4117	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 <_ ( n + 1 ) )
53	34, 41, 37, 48, 52	lemul2ad 9075	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) <_ ( ( ! ` n ) x. ( n + 1 ) ) )
54	35, 38, 42, 46, 53	ltletrd 8558	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. ( n + 1 ) ) )
55		2cnd 9171	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. CC )
56	55, 30	expp1d 10883	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) = ( ( 2 ^ n ) x. 2 ) )
57		facp1 10939	. . . . 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 4114	. . 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 9775	1 |- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4082   ` cfv 5314  (class class class)co 5994   RRcr 7986   1c1 7988   + caddc 7990   x. cmul 7992   < clt 8169   <_ cle 8170   NNcn 9098   2c2 9149   4c4 9151   6c6 9153   NN0cn0 9357  ;cdc 9566   ZZ>=cuz 9710   RR+crp 9837   ^cexp 10747   !cfa 10934

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-mulrcl 8086  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-precex 8097  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103  ax-pre-mulgt0 8104  ax-pre-mulext 8105

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-po 4384  df-iso 4385  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-reap 8710  df-ap 8717  df-div 8808  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-rp 9838  df-seqfrec 10657  df-exp 10748  df-fac 10935

This theorem is referenced by: (None)