Theorem facubnd 9672

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 5198	. . . 4 |- ( m = 0 -> ( ! ` m ) = ( ! ` 0 ) )
2		fac0 9655	. . . 4 |- ( ! ` 0 ) = 1
3	1, 2	syl6eq 2129	. . 3 |- ( m = 0 -> ( ! ` m ) = 1 )
4		id 19	. . . . 5 |- ( m = 0 -> m = 0 )
5	4, 4	oveq12d 5550	. . . 4 |- ( m = 0 -> ( m ^ m ) = ( 0 ^ 0 ) )
6		0exp0e1 9481	. . . 4 |- ( 0 ^ 0 ) = 1
7	5, 6	syl6eq 2129	. . 3 |- ( m = 0 -> ( m ^ m ) = 1 )
8	3, 7	breq12d 3798	. 2 |- ( m = 0 -> ( ( ! ` m ) <_ ( m ^ m ) <-> 1 <_ 1 ) )
9		fveq2 5198	. . 3 |- ( m = k -> ( ! ` m ) = ( ! ` k ) )
10		id 19	. . . 4 |- ( m = k -> m = k )
11	10, 10	oveq12d 5550	. . 3 |- ( m = k -> ( m ^ m ) = ( k ^ k ) )
12	9, 11	breq12d 3798	. 2 |- ( m = k -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` k ) <_ ( k ^ k ) ) )
13		fveq2 5198	. . 3 |- ( m = ( k + 1 ) -> ( ! ` m ) = ( ! ` ( k + 1 ) ) )
14		id 19	. . . 4 |- ( m = ( k + 1 ) -> m = ( k + 1 ) )
15	14, 14	oveq12d 5550	. . 3 |- ( m = ( k + 1 ) -> ( m ^ m ) = ( ( k + 1 ) ^ ( k + 1 ) ) )
16	13, 15	breq12d 3798	. 2 |- ( m = ( k + 1 ) -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
17		fveq2 5198	. . 3 |- ( m = N -> ( ! ` m ) = ( ! ` N ) )
18		id 19	. . . 4 |- ( m = N -> m = N )
19	18, 18	oveq12d 5550	. . 3 |- ( m = N -> ( m ^ m ) = ( N ^ N ) )
20	17, 19	breq12d 3798	. 2 |- ( m = N -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` N ) <_ ( N ^ N ) ) )
21		1le1 7672	. 2 |- 1 <_ 1
22		faccl 9662	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantr 270	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. NN )
24	23	nnred 8052	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. RR )
25		nn0re 8297	. . . . . . . 8 |- ( k e. NN0 -> k e. RR )
26	25	adantr 270	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. RR )
27		simpl 107	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k e. NN0 )
28	26, 27	reexpcld 9622	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) e. RR )
29		nn0p1nn 8327	. . . . . . . . 9 |- ( k e. NN0 -> ( k + 1 ) e. NN )
30	29	adantr 270	. . . . . . . 8 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. NN )
31	30	nnred 8052	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. RR )
32	31, 27	reexpcld 9622	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ k ) e. RR )
33		simpr 108	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( k ^ k ) )
34		nn0ge0 8313	. . . . . . . 8 |- ( k e. NN0 -> 0 <_ k )
35	34	adantr 270	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 <_ k )
36	26	lep1d 8009	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k <_ ( k + 1 ) )
37		leexp1a 9531	. . . . . . 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 1177	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
39	24, 28, 32, 33, 38	letrd 7233	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( ( k + 1 ) ^ k ) )
40	30	nngt0d 8082	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 < ( k + 1 ) )
41		lemul1 7693	. . . . . 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 1173	. . . . 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 145	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( ! ` k ) x. ( k + 1 ) ) <_ ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
44		facp1 9657	. . . . 5 |- ( k e. NN0 -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
45	44	adantr 270	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) = ( ( ! ` k ) x. ( k + 1 ) ) )
46	30	nncnd 8053	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. CC )
47	46, 27	expp1d 9606	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ ( k + 1 ) ) = ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
48	43, 45, 47	3brtr4d 3815	. . 3 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) )
49	48	ex 113	. 2 |- ( k e. NN0 -> ( ( ! ` k ) <_ ( k ^ k ) -> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
50	8, 12, 16, 20, 21, 49	nn0ind 8461	1 |- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   < clt 7153   <_ cle 7154   NNcn 8039   NN0cn0 8288   ^cexp 9475   !cfa 9652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-iseq 9432  df-iexp 9476  df-fac 9653

This theorem is referenced by: (None)