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Theorem inductionexd 36457
Description: Simple induction example. (Contributed by Stanislas Polu, 9-Mar-2020.)
Assertion
Ref Expression
inductionexd  |-  ( N  e.  NN  ->  3  ||  ( ( 4 ^ N )  +  5 ) )

Proof of Theorem inductionexd
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6311 . . . 4  |-  ( k  =  1  ->  (
4 ^ k )  =  ( 4 ^ 1 ) )
21oveq1d 6318 . . 3  |-  ( k  =  1  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ 1 )  +  5 ) )
32breq2d 4433 . 2  |-  ( k  =  1  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ 1 )  +  5 ) ) )
4 oveq2 6311 . . . 4  |-  ( k  =  n  ->  (
4 ^ k )  =  ( 4 ^ n ) )
54oveq1d 6318 . . 3  |-  ( k  =  n  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ n )  +  5 ) )
65breq2d 4433 . 2  |-  ( k  =  n  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ n )  +  5 ) ) )
7 oveq2 6311 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
4 ^ k )  =  ( 4 ^ ( n  +  1 ) ) )
87oveq1d 6318 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ ( n  + 
1 ) )  +  5 ) )
98breq2d 4433 . 2  |-  ( k  =  ( n  + 
1 )  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  5 ) ) )
10 oveq2 6311 . . . 4  |-  ( k  =  N  ->  (
4 ^ k )  =  ( 4 ^ N ) )
1110oveq1d 6318 . . 3  |-  ( k  =  N  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ N )  +  5 ) )
1211breq2d 4433 . 2  |-  ( k  =  N  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ N )  +  5 ) ) )
13 3z 10972 . . . 4  |-  3  e.  ZZ
14 4z 10973 . . . . . 6  |-  4  e.  ZZ
15 1nn0 10887 . . . . . 6  |-  1  e.  NN0
16 zexpcl 12288 . . . . . 6  |-  ( ( 4  e.  ZZ  /\  1  e.  NN0 )  -> 
( 4 ^ 1 )  e.  ZZ )
1714, 15, 16mp2an 677 . . . . 5  |-  ( 4 ^ 1 )  e.  ZZ
18 5nn 10772 . . . . . 6  |-  5  e.  NN
1918nnzi 10963 . . . . 5  |-  5  e.  ZZ
20 zaddcl 10979 . . . . 5  |-  ( ( ( 4 ^ 1 )  e.  ZZ  /\  5  e.  ZZ )  ->  ( ( 4 ^ 1 )  +  5 )  e.  ZZ )
2117, 19, 20mp2an 677 . . . 4  |-  ( ( 4 ^ 1 )  +  5 )  e.  ZZ
2213, 13, 213pm3.2i 1184 . . 3  |-  ( 3  e.  ZZ  /\  3  e.  ZZ  /\  ( ( 4 ^ 1 )  +  5 )  e.  ZZ )
23 3t3e9 10764 . . . 4  |-  ( 3  x.  3 )  =  9
24 4nn0 10890 . . . . . . 7  |-  4  e.  NN0
2524numexp1 15042 . . . . . 6  |-  ( 4 ^ 1 )  =  4
2625oveq1i 6313 . . . . 5  |-  ( ( 4 ^ 1 )  +  5 )  =  ( 4  +  5 )
27 5cn 10691 . . . . . 6  |-  5  e.  CC
28 4cn 10689 . . . . . 6  |-  4  e.  CC
29 5p4e9 10751 . . . . . 6  |-  ( 5  +  4 )  =  9
3027, 28, 29addcomli 9827 . . . . 5  |-  ( 4  +  5 )  =  9
3126, 30eqtri 2452 . . . 4  |-  ( ( 4 ^ 1 )  +  5 )  =  9
3223, 31eqtr4i 2455 . . 3  |-  ( 3  x.  3 )  =  ( ( 4 ^ 1 )  +  5 )
33 dvds0lem 14306 . . 3  |-  ( ( ( 3  e.  ZZ  /\  3  e.  ZZ  /\  ( ( 4 ^ 1 )  +  5 )  e.  ZZ )  /\  ( 3  x.  3 )  =  ( ( 4 ^ 1 )  +  5 ) )  ->  3  ||  ( ( 4 ^ 1 )  +  5 ) )
3422, 32, 33mp2an 677 . 2  |-  3  ||  ( ( 4 ^ 1 )  +  5 )
3513a1i 11 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  e.  ZZ )
36 4nn 10771 . . . . . . . . . . 11  |-  4  e.  NN
3736a1i 11 . . . . . . . . . 10  |-  ( n  e.  NN  ->  4  e.  NN )
38 nnnn0 10878 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
3937, 38nnexpcld 12438 . . . . . . . . 9  |-  ( n  e.  NN  ->  (
4 ^ n )  e.  NN )
4039nnzd 11041 . . . . . . . 8  |-  ( n  e.  NN  ->  (
4 ^ n )  e.  ZZ )
4140adantr 467 . . . . . . 7  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( 4 ^ n
)  e.  ZZ )
4219a1i 11 . . . . . . 7  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
5  e.  ZZ )
4341, 42zaddcld 11046 . . . . . 6  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( ( 4 ^ n )  +  5 )  e.  ZZ )
4414a1i 11 . . . . . 6  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
4  e.  ZZ )
45 simpr 463 . . . . . 6  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
4 ^ n )  +  5 ) )
4635, 43, 44, 45dvdsmultr1d 14332 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
( 4 ^ n
)  +  5 )  x.  4 ) )
47 dvdsmul1 14317 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  5  e.  ZZ )  ->  3  ||  ( 3  x.  5 ) )
4813, 19, 47mp2an 677 . . . . . 6  |-  3  ||  ( 3  x.  5 )
4948a1i 11 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( 3  x.  5 ) )
5043, 44zmulcld 11048 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( ( ( 4 ^ n )  +  5 )  x.  4 )  e.  ZZ )
5135, 42zmulcld 11048 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( 3  x.  5 )  e.  ZZ )
5235, 46, 49, 50, 51dvds2subd 14329 . . . 4  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
5339nncnd 10627 . . . . . . . 8  |-  ( n  e.  NN  ->  (
4 ^ n )  e.  CC )
5427a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  5  e.  CC )
5528a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  4  e.  CC )
5653, 54, 55adddird 9670 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ( 4 ^ n )  +  5 )  x.  4 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) ) )
5756oveq1d 6318 . . . . . 6  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  +  5 )  x.  4 )  - ; 1 5 )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 ) )
58 3cn 10686 . . . . . . . . 9  |-  3  e.  CC
59 5t3e15 11127 . . . . . . . . 9  |-  ( 5  x.  3 )  = ; 1
5
6027, 58, 59mulcomli 9652 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
6160a1i 11 . . . . . . 7  |-  ( n  e.  NN  ->  (
3  x.  5 )  = ; 1 5 )
6261oveq2d 6319 . . . . . 6  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) )  =  ( ( ( ( 4 ^ n
)  +  5 )  x.  4 )  - ; 1 5 ) )
6355, 38expp1d 12418 . . . . . . . 8  |-  ( n  e.  NN  ->  (
4 ^ ( n  +  1 ) )  =  ( ( 4 ^ n )  x.  4 ) )
64 ax-1cn 9599 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
65 3p1e4 10737 . . . . . . . . . . . . . . . 16  |-  ( 3  +  1 )  =  4
6658, 64, 65addcomli 9827 . . . . . . . . . . . . . . 15  |-  ( 1  +  3 )  =  4
6766eqcomi 2436 . . . . . . . . . . . . . 14  |-  4  =  ( 1  +  3 )
6867oveq1i 6313 . . . . . . . . . . . . 13  |-  ( 4  -  3 )  =  ( ( 1  +  3 )  -  3 )
6964, 58pncan3oi 9893 . . . . . . . . . . . . 13  |-  ( ( 1  +  3 )  -  3 )  =  1
7068, 69eqtri 2452 . . . . . . . . . . . 12  |-  ( 4  -  3 )  =  1
7170oveq2i 6314 . . . . . . . . . . 11  |-  ( 5  x.  ( 4  -  3 ) )  =  ( 5  x.  1 )
7227, 28, 58subdii 10069 . . . . . . . . . . 11  |-  ( 5  x.  ( 4  -  3 ) )  =  ( ( 5  x.  4 )  -  (
5  x.  3 ) )
7327mulid1i 9647 . . . . . . . . . . 11  |-  ( 5  x.  1 )  =  5
7471, 72, 733eqtr3ri 2461 . . . . . . . . . 10  |-  5  =  ( ( 5  x.  4 )  -  ( 5  x.  3 ) )
7559eqcomi 2436 . . . . . . . . . . 11  |- ; 1 5  =  ( 5  x.  3 )
7675oveq2i 6314 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  - ; 1 5 )  =  ( ( 5  x.  4 )  -  (
5  x.  3 ) )
7774, 76eqtr4i 2455 . . . . . . . . 9  |-  5  =  ( ( 5  x.  4 )  - ; 1 5 )
7877a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  5  =  ( ( 5  x.  4 )  - ; 1 5 ) )
7963, 78oveq12d 6321 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 5  x.  4 )  - ; 1 5 ) ) )
8053, 55mulcld 9665 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( 4 ^ n
)  x.  4 )  e.  CC )
8154, 55mulcld 9665 . . . . . . . 8  |-  ( n  e.  NN  ->  (
5  x.  4 )  e.  CC )
82 5nn0 10891 . . . . . . . . . . 11  |-  5  e.  NN0
8315, 82deccl 11067 . . . . . . . . . 10  |- ; 1 5  e.  NN0
8483nn0cni 10883 . . . . . . . . 9  |- ; 1 5  e.  CC
8584a1i 11 . . . . . . . 8  |-  ( n  e.  NN  -> ; 1 5  e.  CC )
8680, 81, 85addsubassd 10008 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 5  x.  4 )  - ; 1 5 ) ) )
8779, 86eqtr4d 2467 . . . . . 6  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n
)  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 ) )
8857, 62, 873eqtr4rd 2475 . . . . 5  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n
)  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
8988adantr 467 . . . 4  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( ( 4 ^ ( n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
9052, 89breqtrrd 4448 . . 3  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  5 ) )
9190ex 436 . 2  |-  ( n  e.  NN  ->  (
3  ||  ( (
4 ^ n )  +  5 )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  5 ) ) )
923, 6, 9, 12, 34, 91nnind 10629 1  |-  ( N  e.  NN  ->  3  ||  ( ( 4 ^ N )  +  5 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   class class class wbr 4421  (class class class)co 6303   CCcc 9539   1c1 9542    + caddc 9544    x. cmul 9546    - cmin 9862   NNcn 10611   3c3 10662   4c4 10663   5c5 10664   9c9 10668   NN0cn0 10871   ZZcz 10939  ;cdc 11053   ^cexp 12273    || cdvds 14298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-2 10670  df-3 10671  df-4 10672  df-5 10673  df-6 10674  df-7 10675  df-8 10676  df-9 10677  df-10 10678  df-n0 10872  df-z 10940  df-dec 11054  df-uz 11162  df-seq 12215  df-exp 12274  df-dvds 14299
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator