Theorem 2expltfac 13017

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 6026	. . . 4 |- ( x = 4 -> ( 2 ^ x ) = ( 2 ^ 4 ) )
2		2exp4 13009	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
3	1, 2	eqtrdi 2280	. . 3 |- ( x = 4 -> ( 2 ^ x ) = ; 1 6 )
4		fveq2 5639	. . . 4 |- ( x = 4 -> ( ! ` x ) = ( ! ` 4 ) )
5		fac4 10996	. . . 4 |- ( ! ` 4 ) = ; 2 4
6	4, 5	eqtrdi 2280	. . 3 |- ( x = 4 -> ( ! ` x ) = ; 2 4 )
7	3, 6	breq12d 4101	. 2 |- ( x = 4 -> ( ( 2 ^ x ) < ( ! ` x ) <-> ; 1 6 < ; 2 4 ) )
8		oveq2 6026	. . 3 |- ( x = n -> ( 2 ^ x ) = ( 2 ^ n ) )
9		fveq2 5639	. . 3 |- ( x = n -> ( ! ` x ) = ( ! ` n ) )
10	8, 9	breq12d 4101	. 2 |- ( x = n -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ n ) < ( ! ` n ) ) )
11		oveq2 6026	. . 3 |- ( x = ( n + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( n + 1 ) ) )
12		fveq2 5639	. . 3 |- ( x = ( n + 1 ) -> ( ! ` x ) = ( ! ` ( n + 1 ) ) )
13	11, 12	breq12d 4101	. 2 |- ( x = ( n + 1 ) -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
14		oveq2 6026	. . 3 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
15		fveq2 5639	. . 3 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
16	14, 15	breq12d 4101	. 2 |- ( x = N -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ N ) < ( ! ` N ) ) )
17		1nn0 9418	. . 3 |- 1 e. NN0
18		2nn0 9419	. . 3 |- 2 e. NN0
19		6nn0 9423	. . 3 |- 6 e. NN0
20		4nn0 9421	. . 3 |- 4 e. NN0
21		6lt10 9744	. . 3 |- 6 < ; 1 0
22		1lt2 9313	. . 3 |- 1 < 2
23	17, 18, 19, 20, 21, 22	decltc 9639	. 2 |- ; 1 6 < ; 2 4
24		2nn 9305	. . . . . . . . 9 |- 2 e. NN
25	24	a1i 9	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. NN )
26		4nn 9307	. . . . . . . . . 10 |- 4 e. NN
27		simpl 109	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. ( ZZ>= ` 4 ) )
28		eluznn 9834	. . . . . . . . . 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 9455	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN0 )
31	25, 30	nnexpcld 10958	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. NN )
32	31	nnred 9156	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. RR )
33		2re 9213	. . . . . . 7 |- 2 e. RR
34	33	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR )
35	32, 34	remulcld 8210	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) e. RR )
36	30	faccld 10999	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN )
37	36	nnred 9156	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. RR )
38	37, 34	remulcld 8210	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) e. RR )
39	29	nnred 9156	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. RR )
40		1red 8194	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 e. RR )
41	39, 40	readdcld 8209	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( n + 1 ) e. RR )
42	37, 41	remulcld 8210	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. ( n + 1 ) ) e. RR )
43		2rp 9893	. . . . . . 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 9987	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. 2 ) )
47	36	nnnn0d 9455	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN0 )
48	47	nn0ge0d 9458	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 0 <_ ( ! ` n ) )
49		df-2 9202	. . . . . . 7 |- 2 = ( 1 + 1 )
50	29	nnge1d 9186	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 <_ n )
51	40, 39, 40, 50	leadd1dd 8739	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 1 + 1 ) <_ ( n + 1 ) )
52	49, 51	eqbrtrid 4123	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 <_ ( n + 1 ) )
53	34, 41, 37, 48, 52	lemul2ad 9120	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) <_ ( ( ! ` n ) x. ( n + 1 ) ) )
54	35, 38, 42, 46, 53	ltletrd 8603	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. ( n + 1 ) ) )
55		2cnd 9216	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. CC )
56	55, 30	expp1d 10937	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) = ( ( 2 ^ n ) x. 2 ) )
57		facp1 10993	. . . . 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 4120	. . 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 9826	1 |- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   RRcr 8031   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   <_ cle 8215   NNcn 9143   2c2 9194   4c4 9196   6c6 9198   NN0cn0 9402  ;cdc 9611   ZZ>=cuz 9755   RR+crp 9888   ^cexp 10801   !cfa 10988

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-z 9480  df-dec 9612  df-uz 9756  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-fac 10989

This theorem is referenced by: (None)