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Theorem faclbnd4lem3 12376
Description: Lemma for faclbnd4 12378. The  N  =  0 case. (Contributed by NM, 23-Dec-2005.)
Assertion
Ref Expression
faclbnd4lem3  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( ( N ^ K )  x.  ( M ^ N
) )  <_  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) ) )

Proof of Theorem faclbnd4lem3
StepHypRef Expression
1 elnn0 10818 . . . . 5  |-  ( K  e.  NN0  <->  ( K  e.  NN  \/  K  =  0 ) )
2 0exp 12204 . . . . . . . 8  |-  ( K  e.  NN  ->  (
0 ^ K )  =  0 )
32adantl 466 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  =  0 )
4 nnnn0 10823 . . . . . . . . 9  |-  ( K  e.  NN  ->  K  e.  NN0 )
5 2nn0 10833 . . . . . . . . . . . 12  |-  2  e.  NN0
6 nn0sqcl 12196 . . . . . . . . . . . 12  |-  ( K  e.  NN0  ->  ( K ^ 2 )  e. 
NN0 )
7 nn0expcl 12183 . . . . . . . . . . . 12  |-  ( ( 2  e.  NN0  /\  ( K ^ 2 )  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
85, 6, 7sylancr 663 . . . . . . . . . . 11  |-  ( K  e.  NN0  ->  ( 2 ^ ( K ^
2 ) )  e. 
NN0 )
98adantl 466 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 2 ^ ( K ^ 2 ) )  e.  NN0 )
10 nn0addcl 10852 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M  +  K
)  e.  NN0 )
11 nn0expcl 12183 . . . . . . . . . . 11  |-  ( ( M  e.  NN0  /\  ( M  +  K
)  e.  NN0 )  ->  ( M ^ ( M  +  K )
)  e.  NN0 )
1210, 11syldan 470 . . . . . . . . . 10  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( M ^ ( M  +  K )
)  e.  NN0 )
139, 12nn0mulcld 10878 . . . . . . . . 9  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
144, 13sylan2 474 . . . . . . . 8  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  NN0 )
1514nn0ge0d 10876 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  0  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
163, 15eqbrtrd 4476 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  e.  NN )  ->  ( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
17 1nn 10567 . . . . . . . . . 10  |-  1  e.  NN
18 elnn0 10818 . . . . . . . . . . 11  |-  ( M  e.  NN0  <->  ( M  e.  NN  \/  M  =  0 ) )
19 nnnn0 10823 . . . . . . . . . . . . . 14  |-  ( M  e.  NN  ->  M  e.  NN0 )
20 0nn0 10831 . . . . . . . . . . . . . 14  |-  0  e.  NN0
21 nn0addcl 10852 . . . . . . . . . . . . . 14  |-  ( ( M  e.  NN0  /\  0  e.  NN0 )  -> 
( M  +  0 )  e.  NN0 )
2219, 20, 21sylancl 662 . . . . . . . . . . . . 13  |-  ( M  e.  NN  ->  ( M  +  0 )  e.  NN0 )
23 nnexpcl 12182 . . . . . . . . . . . . 13  |-  ( ( M  e.  NN  /\  ( M  +  0
)  e.  NN0 )  ->  ( M ^ ( M  +  0 ) )  e.  NN )
2422, 23mpdan 668 . . . . . . . . . . . 12  |-  ( M  e.  NN  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
25 id 22 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  M  =  0 )
26 oveq1 6303 . . . . . . . . . . . . . . . 16  |-  ( M  =  0  ->  ( M  +  0 )  =  ( 0  +  0 ) )
27 00id 9772 . . . . . . . . . . . . . . . 16  |-  ( 0  +  0 )  =  0
2826, 27syl6eq 2514 . . . . . . . . . . . . . . 15  |-  ( M  =  0  ->  ( M  +  0 )  =  0 )
2925, 28oveq12d 6314 . . . . . . . . . . . . . 14  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  ( 0 ^ 0 ) )
30 0exp0e1 12174 . . . . . . . . . . . . . 14  |-  ( 0 ^ 0 )  =  1
3129, 30syl6eq 2514 . . . . . . . . . . . . 13  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  =  1 )
3231, 17syl6eqel 2553 . . . . . . . . . . . 12  |-  ( M  =  0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
3324, 32jaoi 379 . . . . . . . . . . 11  |-  ( ( M  e.  NN  \/  M  =  0 )  ->  ( M ^
( M  +  0 ) )  e.  NN )
3418, 33sylbi 195 . . . . . . . . . 10  |-  ( M  e.  NN0  ->  ( M ^ ( M  + 
0 ) )  e.  NN )
35 nnmulcl 10579 . . . . . . . . . 10  |-  ( ( 1  e.  NN  /\  ( M ^ ( M  +  0 ) )  e.  NN )  -> 
( 1  x.  ( M ^ ( M  + 
0 ) ) )  e.  NN )
3617, 34, 35sylancr 663 . . . . . . . . 9  |-  ( M  e.  NN0  ->  ( 1  x.  ( M ^
( M  +  0 ) ) )  e.  NN )
3736nnge1d 10599 . . . . . . . 8  |-  ( M  e.  NN0  ->  1  <_ 
( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
3837adantr 465 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  1  <_  (
1  x.  ( M ^ ( M  + 
0 ) ) ) )
39 oveq2 6304 . . . . . . . . . 10  |-  ( K  =  0  ->  (
0 ^ K )  =  ( 0 ^ 0 ) )
4039, 30syl6eq 2514 . . . . . . . . 9  |-  ( K  =  0  ->  (
0 ^ K )  =  1 )
41 sq0i 12263 . . . . . . . . . . . 12  |-  ( K  =  0  ->  ( K ^ 2 )  =  0 )
4241oveq2d 6312 . . . . . . . . . . 11  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  ( 2 ^ 0 ) )
43 2cn 10627 . . . . . . . . . . . 12  |-  2  e.  CC
44 exp0 12173 . . . . . . . . . . . 12  |-  ( 2  e.  CC  ->  (
2 ^ 0 )  =  1 )
4543, 44ax-mp 5 . . . . . . . . . . 11  |-  ( 2 ^ 0 )  =  1
4642, 45syl6eq 2514 . . . . . . . . . 10  |-  ( K  =  0  ->  (
2 ^ ( K ^ 2 ) )  =  1 )
47 oveq2 6304 . . . . . . . . . . 11  |-  ( K  =  0  ->  ( M  +  K )  =  ( M  + 
0 ) )
4847oveq2d 6312 . . . . . . . . . 10  |-  ( K  =  0  ->  ( M ^ ( M  +  K ) )  =  ( M ^ ( M  +  0 ) ) )
4946, 48oveq12d 6314 . . . . . . . . 9  |-  ( K  =  0  ->  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  =  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) )
5040, 49breq12d 4469 . . . . . . . 8  |-  ( K  =  0  ->  (
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  <->  1  <_  ( 1  x.  ( M ^ ( M  + 
0 ) ) ) ) )
5150adantl 466 . . . . . . 7  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( ( 0 ^ K )  <_ 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  <->  1  <_  ( 1  x.  ( M ^
( M  +  0 ) ) ) ) )
5238, 51mpbird 232 . . . . . 6  |-  ( ( M  e.  NN0  /\  K  =  0 )  ->  ( 0 ^ K )  <_  (
( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
5316, 52jaodan 785 . . . . 5  |-  ( ( M  e.  NN0  /\  ( K  e.  NN  \/  K  =  0
) )  ->  (
0 ^ K )  <_  ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) ) )
541, 53sylan2b 475 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  <_  ( (
2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
55 nn0cn 10826 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  CC )
5655exp0d 12307 . . . . . 6  |-  ( M  e.  NN0  ->  ( M ^ 0 )  =  1 )
5756oveq2d 6312 . . . . 5  |-  ( M  e.  NN0  ->  ( ( 0 ^ K )  x.  ( M ^
0 ) )  =  ( ( 0 ^ K )  x.  1 ) )
58 nn0expcl 12183 . . . . . . . 8  |-  ( ( 0  e.  NN0  /\  K  e.  NN0 )  -> 
( 0 ^ K
)  e.  NN0 )
5920, 58mpan 670 . . . . . . 7  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e. 
NN0 )
6059nn0cnd 10875 . . . . . 6  |-  ( K  e.  NN0  ->  ( 0 ^ K )  e.  CC )
6160mulid1d 9630 . . . . 5  |-  ( K  e.  NN0  ->  ( ( 0 ^ K )  x.  1 )  =  ( 0 ^ K
) )
6257, 61sylan9eq 2518 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  =  ( 0 ^ K ) )
6313nn0cnd 10875 . . . . 5  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  e.  CC )
6463mulid1d 9630 . . . 4  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 )  =  ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) ) )
6554, 62, 643brtr4d 4486 . . 3  |-  ( ( M  e.  NN0  /\  K  e.  NN0 )  -> 
( ( 0 ^ K )  x.  ( M ^ 0 ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  1 ) )
6665adantr 465 . 2  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
67 oveq1 6303 . . . . 5  |-  ( N  =  0  ->  ( N ^ K )  =  ( 0 ^ K
) )
68 oveq2 6304 . . . . 5  |-  ( N  =  0  ->  ( M ^ N )  =  ( M ^ 0 ) )
6967, 68oveq12d 6314 . . . 4  |-  ( N  =  0  ->  (
( N ^ K
)  x.  ( M ^ N ) )  =  ( ( 0 ^ K )  x.  ( M ^ 0 ) ) )
70 fveq2 5872 . . . . . 6  |-  ( N  =  0  ->  ( ! `  N )  =  ( ! ` 
0 ) )
71 fac0 12359 . . . . . 6  |-  ( ! `
 0 )  =  1
7270, 71syl6eq 2514 . . . . 5  |-  ( N  =  0  ->  ( ! `  N )  =  1 )
7372oveq2d 6312 . . . 4  |-  ( N  =  0  ->  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) )  =  ( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) )
7469, 73breq12d 4469 . . 3  |-  ( N  =  0  ->  (
( ( N ^ K )  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  ( ! `  N
) )  <->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) ) )
7574adantl 466 . 2  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( (
( N ^ K
)  x.  ( M ^ N ) )  <_  ( ( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^
( M  +  K
) ) )  x.  ( ! `  N
) )  <->  ( (
0 ^ K )  x.  ( M ^
0 ) )  <_ 
( ( ( 2 ^ ( K ^
2 ) )  x.  ( M ^ ( M  +  K )
) )  x.  1 ) ) )
7666, 75mpbird 232 1  |-  ( ( ( M  e.  NN0  /\  K  e.  NN0 )  /\  N  =  0
)  ->  ( ( N ^ K )  x.  ( M ^ N
) )  <_  (
( ( 2 ^ ( K ^ 2 ) )  x.  ( M ^ ( M  +  K ) ) )  x.  ( ! `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    <_ cle 9646   NNcn 10556   2c2 10606   NN0cn0 10816   ^cexp 12169   !cfa 12356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12111  df-exp 12170  df-fac 12357
This theorem is referenced by:  faclbnd4lem4  12377  faclbnd4  12378
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