Theorem 2expltfac 13073

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 6036	. . . 4 |- ( x = 4 -> ( 2 ^ x ) = ( 2 ^ 4 ) )
2		2exp4 13065	. . . 4 |- ( 2 ^ 4 ) = ; 1 6
3	1, 2	eqtrdi 2280	. . 3 |- ( x = 4 -> ( 2 ^ x ) = ; 1 6 )
4		fveq2 5648	. . . 4 |- ( x = 4 -> ( ! ` x ) = ( ! ` 4 ) )
5		fac4 11039	. . . 4 |- ( ! ` 4 ) = ; 2 4
6	4, 5	eqtrdi 2280	. . 3 |- ( x = 4 -> ( ! ` x ) = ; 2 4 )
7	3, 6	breq12d 4106	. 2 |- ( x = 4 -> ( ( 2 ^ x ) < ( ! ` x ) <-> ; 1 6 < ; 2 4 ) )
8		oveq2 6036	. . 3 |- ( x = n -> ( 2 ^ x ) = ( 2 ^ n ) )
9		fveq2 5648	. . 3 |- ( x = n -> ( ! ` x ) = ( ! ` n ) )
10	8, 9	breq12d 4106	. 2 |- ( x = n -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ n ) < ( ! ` n ) ) )
11		oveq2 6036	. . 3 |- ( x = ( n + 1 ) -> ( 2 ^ x ) = ( 2 ^ ( n + 1 ) ) )
12		fveq2 5648	. . 3 |- ( x = ( n + 1 ) -> ( ! ` x ) = ( ! ` ( n + 1 ) ) )
13	11, 12	breq12d 4106	. 2 |- ( x = ( n + 1 ) -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ ( n + 1 ) ) < ( ! ` ( n + 1 ) ) ) )
14		oveq2 6036	. . 3 |- ( x = N -> ( 2 ^ x ) = ( 2 ^ N ) )
15		fveq2 5648	. . 3 |- ( x = N -> ( ! ` x ) = ( ! ` N ) )
16	14, 15	breq12d 4106	. 2 |- ( x = N -> ( ( 2 ^ x ) < ( ! ` x ) <-> ( 2 ^ N ) < ( ! ` N ) ) )
17		1nn0 9461	. . 3 |- 1 e. NN0
18		2nn0 9462	. . 3 |- 2 e. NN0
19		6nn0 9466	. . 3 |- 6 e. NN0
20		4nn0 9464	. . 3 |- 4 e. NN0
21		6lt10 9787	. . 3 |- 6 < ; 1 0
22		1lt2 9356	. . 3 |- 1 < 2
23	17, 18, 19, 20, 21, 22	decltc 9682	. 2 |- ; 1 6 < ; 2 4
24		2nn 9348	. . . . . . . . 9 |- 2 e. NN
25	24	a1i 9	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. NN )
26		4nn 9350	. . . . . . . . . 10 |- 4 e. NN
27		simpl 109	. . . . . . . . . 10 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. ( ZZ>= ` 4 ) )
28		eluznn 9877	. . . . . . . . . 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 9498	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. NN0 )
31	25, 30	nnexpcld 11001	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. NN )
32	31	nnred 9199	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ n ) e. RR )
33		2re 9256	. . . . . . 7 |- 2 e. RR
34	33	a1i 9	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. RR )
35	32, 34	remulcld 8253	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) e. RR )
36	30	faccld 11042	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN )
37	36	nnred 9199	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. RR )
38	37, 34	remulcld 8253	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) e. RR )
39	29	nnred 9199	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> n e. RR )
40		1red 8237	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 e. RR )
41	39, 40	readdcld 8252	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( n + 1 ) e. RR )
42	37, 41	remulcld 8253	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. ( n + 1 ) ) e. RR )
43		2rp 9936	. . . . . . 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 10030	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. 2 ) )
47	36	nnnn0d 9498	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ! ` n ) e. NN0 )
48	47	nn0ge0d 9501	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 0 <_ ( ! ` n ) )
49		df-2 9245	. . . . . . 7 |- 2 = ( 1 + 1 )
50	29	nnge1d 9229	. . . . . . . 8 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 1 <_ n )
51	40, 39, 40, 50	leadd1dd 8782	. . . . . . 7 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 1 + 1 ) <_ ( n + 1 ) )
52	49, 51	eqbrtrid 4128	. . . . . 6 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 <_ ( n + 1 ) )
53	34, 41, 37, 48, 52	lemul2ad 9163	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( ! ` n ) x. 2 ) <_ ( ( ! ` n ) x. ( n + 1 ) ) )
54	35, 38, 42, 46, 53	ltletrd 8646	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( ( 2 ^ n ) x. 2 ) < ( ( ! ` n ) x. ( n + 1 ) ) )
55		2cnd 9259	. . . . 5 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> 2 e. CC )
56	55, 30	expp1d 10980	. . . 4 |- ( ( n e. ( ZZ>= ` 4 ) /\ ( 2 ^ n ) < ( ! ` n ) ) -> ( 2 ^ ( n + 1 ) ) = ( ( 2 ^ n ) x. 2 ) )
57		facp1 11036	. . . . 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 4125	. . 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 9869	1 |- ( N e. ( ZZ>= ` 4 ) -> ( 2 ^ N ) < ( ! ` N ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   RRcr 8074   1c1 8076   + caddc 8078   x. cmul 8080   < clt 8257   <_ cle 8258   NNcn 9186   2c2 9237   4c4 9239   6c6 9241   NN0cn0 9445  ;cdc 9654   ZZ>=cuz 9798   RR+crp 9931   ^cexp 10844   !cfa 11031

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-reap 8798  df-ap 8805  df-div 8896  df-inn 9187  df-2 9245  df-3 9246  df-4 9247  df-5 9248  df-6 9249  df-7 9250  df-8 9251  df-9 9252  df-n0 9446  df-z 9523  df-dec 9655  df-uz 9799  df-rp 9932  df-seqfrec 10754  df-exp 10845  df-fac 11032

This theorem is referenced by: (None)