Theorem facubnd 11006

Description: An upper bound for the factorial function. (Contributed by Mario Carneiro, 15-Apr-2016.)

Assertion

Ref	Expression
facubnd	|- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Proof of Theorem facubnd

Dummy variables m k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5639	. . . 4 |- ( m = 0 -> ( ! ` m ) = ( ! ` 0 ) )
2		fac0 10989	. . . 4 |- ( ! ` 0 ) = 1
3	1, 2	eqtrdi 2280	. . 3 |- ( m = 0 -> ( ! ` m ) = 1 )
4		id 19	. . . . 5 |- ( m = 0 -> m = 0 )
5	4, 4	oveq12d 6035	. . . 4 |- ( m = 0 -> ( m ^ m ) = ( 0 ^ 0 ) )
6		0exp0e1 10805	. . . 4 |- ( 0 ^ 0 ) = 1
7	5, 6	eqtrdi 2280	. . 3 |- ( m = 0 -> ( m ^ m ) = 1 )
8	3, 7	breq12d 4101	. 2 |- ( m = 0 -> ( ( ! ` m ) <_ ( m ^ m ) <-> 1 <_ 1 ) )
9		fveq2 5639	. . 3 |- ( m = k -> ( ! ` m ) = ( ! ` k ) )
10		id 19	. . . 4 |- ( m = k -> m = k )
11	10, 10	oveq12d 6035	. . 3 |- ( m = k -> ( m ^ m ) = ( k ^ k ) )
12	9, 11	breq12d 4101	. 2 |- ( m = k -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` k ) <_ ( k ^ k ) ) )
13		fveq2 5639	. . 3 |- ( m = ( k + 1 ) -> ( ! ` m ) = ( ! ` ( k + 1 ) ) )
14		id 19	. . . 4 |- ( m = ( k + 1 ) -> m = ( k + 1 ) )
15	14, 14	oveq12d 6035	. . 3 |- ( m = ( k + 1 ) -> ( m ^ m ) = ( ( k + 1 ) ^ ( k + 1 ) ) )
16	13, 15	breq12d 4101	. 2 |- ( m = ( k + 1 ) -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
17		fveq2 5639	. . 3 |- ( m = N -> ( ! ` m ) = ( ! ` N ) )
18		id 19	. . . 4 |- ( m = N -> m = N )
19	18, 18	oveq12d 6035	. . 3 |- ( m = N -> ( m ^ m ) = ( N ^ N ) )
20	17, 19	breq12d 4101	. 2 |- ( m = N -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` N ) <_ ( N ^ N ) ) )
21		1le1 8751	. 2 |- 1 <_ 1
22		faccl 10996	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantr 276	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. NN )
24	23	nnred 9155	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. RR )
25		nn0re 9410	. . . . . . . 8 |- ( k e. NN0 -> k e. RR )
26	25	adantr 276	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. RR )
27		simpl 109	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. NN0 )
28	26, 27	reexpcld 10951	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) e. RR )
29		nn0p1nn 9440	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN )
30	29	adantr 276	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. NN )
31	30	nnred 9155	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. RR )
32	31, 27	reexpcld 10951	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ k ) e. RR )
33		simpr 110	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( k ^ k ) )
34		nn0ge0 9426	. . . . . . . 8 |- ( k e. NN0 -> 0 <_ k )
35	34	adantr 276	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 <_ k )
36	26	lep1d 9110	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k <_ ( k + 1 ) )
37		leexp1a 10855	. . . . . . 7 |- ( ( ( k e. RR /\ ( k + 1 ) e. RR /\ k e. NN0 ) /\ ( 0 <_ k /\ k <_ ( k + 1 ) ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
38	26, 31, 27, 35, 36, 37	syl32anc 1281	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
39	24, 28, 32, 33, 38	letrd 8302	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( ( k + 1 ) ^ k ) )
40	30	nngt0d 9186	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 < ( k + 1 ) )
41		lemul1 8772	. . . . . 6 |- ( ( ( ! ` k ) e. RR /\ ( ( k + 1 ) ^ k ) e. RR /\ ( ( k + 1 ) e. RR /\ 0 < ( k + 1 ) ) ) -> ( ( ! ` k ) <_ ( ( k + 1 ) ^ k ) <-> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) ) )
42	24, 32, 31, 40, 41	syl112anc 1277	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( ! ` k ) <_ ( ( k + 1 ) ^ k ) <-> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) ) )
43	39, 42	mpbid 147	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
44		facp1 10991	. . . . 5 |- ( k e. NN0 -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
45	44	adantr 276	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
46	30	nncnd 9156	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. CC )
47	46, 27	expp1d 10935	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ ( k + 1 ) ) = ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
48	43, 45, 47	3brtr4d 4120	. . 3 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) )
49	48	ex 115	. 2 |- ( k e. NN0 -> ( ( ! ` k ) <_ ( k ^ k ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
50	8, 12, 16, 20, 21, 49	nn0ind 9593	1 |- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   NNcn 9142   NN0cn0 9401   ^cexp 10799   !cfa 10986

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-seqfrec 10709  df-exp 10800  df-fac 10987

This theorem is referenced by: (None)