Theorem facubnd 11161

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 5690	. . . 4 |- ( m = 0 -> ( ! ` m ) = ( ! ` 0 ) )
2		fac0 11144	. . . 4 |- ( ! ` 0 ) = 1
3	1, 2	eqtrdi 2287	. . 3 |- ( m = 0 -> ( ! ` m ) = 1 )
4		id 19	. . . . 5 |- ( m = 0 -> m = 0 )
5	4, 4	oveq12d 6093	. . . 4 |- ( m = 0 -> ( m ^ m ) = ( 0 ^ 0 ) )
6		0exp0e1 10959	. . . 4 |- ( 0 ^ 0 ) = 1
7	5, 6	eqtrdi 2287	. . 3 |- ( m = 0 -> ( m ^ m ) = 1 )
8	3, 7	breq12d 4138	. 2 |- ( m = 0 -> ( ( ! ` m ) <_ ( m ^ m ) <-> 1 <_ 1 ) )
9		fveq2 5690	. . 3 |- ( m = k -> ( ! ` m ) = ( ! ` k ) )
10		id 19	. . . 4 |- ( m = k -> m = k )
11	10, 10	oveq12d 6093	. . 3 |- ( m = k -> ( m ^ m ) = ( k ^ k ) )
12	9, 11	breq12d 4138	. 2 |- ( m = k -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` k ) <_ ( k ^ k ) ) )
13		fveq2 5690	. . 3 |- ( m = ( k + 1 ) -> ( ! ` m ) = ( ! ` ( k + 1 ) ) )
14		id 19	. . . 4 |- ( m = ( k + 1 ) -> m = ( k + 1 ) )
15	14, 14	oveq12d 6093	. . 3 |- ( m = ( k + 1 ) -> ( m ^ m ) = ( ( k + 1 ) ^ ( k + 1 ) ) )
16	13, 15	breq12d 4138	. 2 |- ( m = ( k + 1 ) -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` ( k + 1 ) ) <_ ( ( k + 1 ) ^ ( k + 1 ) ) ) )
17		fveq2 5690	. . 3 |- ( m = N -> ( ! ` m ) = ( ! ` N ) )
18		id 19	. . . 4 |- ( m = N -> m = N )
19	18, 18	oveq12d 6093	. . 3 |- ( m = N -> ( m ^ m ) = ( N ^ N ) )
20	17, 19	breq12d 4138	. 2 |- ( m = N -> ( ( ! ` m ) <_ ( m ^ m ) <-> ( ! ` N ) <_ ( N ^ N ) ) )
21		1le1 8890	. 2 |- 1 <_ 1
22		faccl 11151	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
23	22	adantr 276	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. NN )
24	23	nnred 9296	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) e. RR )
25		nn0re 9551	. . . . . . . 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 11106	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) e. RR )
29		nn0p1nn 9581	. . . . . . . . 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 9296	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. RR )
32	31, 27	reexpcld 11106	. . . . . 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 9567	. . . . . . . 8 |- ( k e. NN0 -> 0 <_ k )
35	34	adantr 276	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 <_ k )
36	26	lep1d 9251	. . . . . . 7 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> k <_ ( k + 1 ) )
37		leexp1a 11009	. . . . . . 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 1286	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k ^ k ) <_ ( ( k + 1 ) ^ k ) )
39	24, 28, 32, 33, 38	letrd 8440	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ! ` k ) <_ ( ( k + 1 ) ^ k ) )
40	30	nngt0d 9327	. . . . . 6 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> 0 < ( k + 1 ) )
41		lemul1 8911	. . . . . 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 1282	. . . . 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 11146	. . . . 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 9297	. . . . 5 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( k + 1 ) e. CC )
47	46, 27	expp1d 11090	. . . 4 |- ( ( k e. NN0 /\ ( ! ` k ) <_ ( k ^ k ) ) -> ( ( k + 1 ) ^ ( k + 1 ) ) = ( ( ( k + 1 ) ^ k ) x. ( k + 1 ) ) )
48	43, 45, 47	3brtr4d 4157	. . 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 9739	1 |- ( N e. NN0 -> ( ! ` N ) <_ ( N ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   NNcn 9283   NN0cn0 9542   ^cexp 10953   !cfa 11141

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863  df-exp 10954  df-fac 11142

This theorem is referenced by: (None)