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Theorem inductionexd 38177
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 6304 . . . 4  |-  ( k  =  1  ->  (
4 ^ k )  =  ( 4 ^ 1 ) )
21oveq1d 6311 . . 3  |-  ( k  =  1  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ 1 )  +  5 ) )
32breq2d 4468 . 2  |-  ( k  =  1  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ 1 )  +  5 ) ) )
4 oveq2 6304 . . . 4  |-  ( k  =  n  ->  (
4 ^ k )  =  ( 4 ^ n ) )
54oveq1d 6311 . . 3  |-  ( k  =  n  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ n )  +  5 ) )
65breq2d 4468 . 2  |-  ( k  =  n  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ n )  +  5 ) ) )
7 oveq2 6304 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
4 ^ k )  =  ( 4 ^ ( n  +  1 ) ) )
87oveq1d 6311 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ ( n  + 
1 ) )  +  5 ) )
98breq2d 4468 . 2  |-  ( k  =  ( n  + 
1 )  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  5 ) ) )
10 oveq2 6304 . . . 4  |-  ( k  =  N  ->  (
4 ^ k )  =  ( 4 ^ N ) )
1110oveq1d 6311 . . 3  |-  ( k  =  N  ->  (
( 4 ^ k
)  +  5 )  =  ( ( 4 ^ N )  +  5 ) )
1211breq2d 4468 . 2  |-  ( k  =  N  ->  (
3  ||  ( (
4 ^ k )  +  5 )  <->  3  ||  ( ( 4 ^ N )  +  5 ) ) )
13 3z 10918 . . . 4  |-  3  e.  ZZ
14 4z 10919 . . . . . 6  |-  4  e.  ZZ
15 1nn0 10832 . . . . . 6  |-  1  e.  NN0
16 zexpcl 12184 . . . . . 6  |-  ( ( 4  e.  ZZ  /\  1  e.  NN0 )  -> 
( 4 ^ 1 )  e.  ZZ )
1714, 15, 16mp2an 672 . . . . 5  |-  ( 4 ^ 1 )  e.  ZZ
18 5nn 10717 . . . . . 6  |-  5  e.  NN
1918nnzi 10909 . . . . 5  |-  5  e.  ZZ
20 zaddcl 10925 . . . . 5  |-  ( ( ( 4 ^ 1 )  e.  ZZ  /\  5  e.  ZZ )  ->  ( ( 4 ^ 1 )  +  5 )  e.  ZZ )
2117, 19, 20mp2an 672 . . . 4  |-  ( ( 4 ^ 1 )  +  5 )  e.  ZZ
2213, 13, 213pm3.2i 1174 . . 3  |-  ( 3  e.  ZZ  /\  3  e.  ZZ  /\  ( ( 4 ^ 1 )  +  5 )  e.  ZZ )
23 3t3e9 10709 . . . 4  |-  ( 3  x.  3 )  =  9
24 4nn0 10835 . . . . . . 7  |-  4  e.  NN0
2524numexp1 14666 . . . . . 6  |-  ( 4 ^ 1 )  =  4
2625oveq1i 6306 . . . . 5  |-  ( ( 4 ^ 1 )  +  5 )  =  ( 4  +  5 )
27 5cn 10636 . . . . . 6  |-  5  e.  CC
28 4cn 10634 . . . . . 6  |-  4  e.  CC
29 5p4e9 10696 . . . . . 6  |-  ( 5  +  4 )  =  9
3027, 28, 29addcomli 9789 . . . . 5  |-  ( 4  +  5 )  =  9
3126, 30eqtri 2486 . . . 4  |-  ( ( 4 ^ 1 )  +  5 )  =  9
3223, 31eqtr4i 2489 . . 3  |-  ( 3  x.  3 )  =  ( ( 4 ^ 1 )  +  5 )
33 dvds0lem 14097 . . 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 672 . 2  |-  3  ||  ( ( 4 ^ 1 )  +  5 )
3513a1i 11 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  e.  ZZ )
36 4nn 10716 . . . . . . . . . . 11  |-  4  e.  NN
3736a1i 11 . . . . . . . . . 10  |-  ( n  e.  NN  ->  4  e.  NN )
38 nnnn0 10823 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  NN0 )
3937, 38nnexpcld 12334 . . . . . . . . 9  |-  ( n  e.  NN  ->  (
4 ^ n )  e.  NN )
4039nnzd 10989 . . . . . . . 8  |-  ( n  e.  NN  ->  (
4 ^ n )  e.  ZZ )
4140adantr 465 . . . . . . 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 10994 . . . . . 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 461 . . . . . 6  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
4 ^ n )  +  5 ) )
4635, 43, 44, 45dvdsmultr1d 14123 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
( 4 ^ n
)  +  5 )  x.  4 ) )
47 dvdsmul1 14108 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  5  e.  ZZ )  ->  3  ||  ( 3  x.  5 ) )
4813, 19, 47mp2an 672 . . . . . 6  |-  3  ||  ( 3  x.  5 )
4948a1i 11 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( 3  x.  5 ) )
5043, 44zmulcld 10996 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( ( ( 4 ^ n )  +  5 )  x.  4 )  e.  ZZ )
5135, 42zmulcld 10996 . . . . 5  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( 3  x.  5 )  e.  ZZ )
5235, 46, 49, 50, 51dvds2subd 14120 . . . 4  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
5339nncnd 10572 . . . . . . . 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 9638 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ( 4 ^ n )  +  5 )  x.  4 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) ) )
5756oveq1d 6311 . . . . . 6  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  +  5 )  x.  4 )  - ; 1 5 )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 ) )
58 3cn 10631 . . . . . . . . 9  |-  3  e.  CC
59 5t3e15 11074 . . . . . . . . 9  |-  ( 5  x.  3 )  = ; 1
5
6027, 58, 59mulcomli 9620 . . . . . . . 8  |-  ( 3  x.  5 )  = ; 1
5
6160a1i 11 . . . . . . 7  |-  ( n  e.  NN  ->  (
3  x.  5 )  = ; 1 5 )
6261oveq2d 6312 . . . . . 6  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) )  =  ( ( ( ( 4 ^ n
)  +  5 )  x.  4 )  - ; 1 5 ) )
6355, 38expp1d 12314 . . . . . . . 8  |-  ( n  e.  NN  ->  (
4 ^ ( n  +  1 ) )  =  ( ( 4 ^ n )  x.  4 ) )
64 ax-1cn 9567 . . . . . . . . . . . . . . . 16  |-  1  e.  CC
65 3p1e4 10682 . . . . . . . . . . . . . . . 16  |-  ( 3  +  1 )  =  4
6658, 64, 65addcomli 9789 . . . . . . . . . . . . . . 15  |-  ( 1  +  3 )  =  4
6766eqcomi 2470 . . . . . . . . . . . . . 14  |-  4  =  ( 1  +  3 )
6867oveq1i 6306 . . . . . . . . . . . . 13  |-  ( 4  -  3 )  =  ( ( 1  +  3 )  -  3 )
6964, 58pncan3oi 9855 . . . . . . . . . . . . 13  |-  ( ( 1  +  3 )  -  3 )  =  1
7068, 69eqtri 2486 . . . . . . . . . . . 12  |-  ( 4  -  3 )  =  1
7170oveq2i 6307 . . . . . . . . . . 11  |-  ( 5  x.  ( 4  -  3 ) )  =  ( 5  x.  1 )
7227, 28, 58subdii 10026 . . . . . . . . . . 11  |-  ( 5  x.  ( 4  -  3 ) )  =  ( ( 5  x.  4 )  -  (
5  x.  3 ) )
7327mulid1i 9615 . . . . . . . . . . 11  |-  ( 5  x.  1 )  =  5
7471, 72, 733eqtr3ri 2495 . . . . . . . . . 10  |-  5  =  ( ( 5  x.  4 )  -  ( 5  x.  3 ) )
7559eqcomi 2470 . . . . . . . . . . 11  |- ; 1 5  =  ( 5  x.  3 )
7675oveq2i 6307 . . . . . . . . . 10  |-  ( ( 5  x.  4 )  - ; 1 5 )  =  ( ( 5  x.  4 )  -  (
5  x.  3 ) )
7774, 76eqtr4i 2489 . . . . . . . . 9  |-  5  =  ( ( 5  x.  4 )  - ; 1 5 )
7877a1i 11 . . . . . . . 8  |-  ( n  e.  NN  ->  5  =  ( ( 5  x.  4 )  - ; 1 5 ) )
7963, 78oveq12d 6314 . . . . . . 7  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 5  x.  4 )  - ; 1 5 ) ) )
8053, 55mulcld 9633 . . . . . . . 8  |-  ( n  e.  NN  ->  (
( 4 ^ n
)  x.  4 )  e.  CC )
8154, 55mulcld 9633 . . . . . . . 8  |-  ( n  e.  NN  ->  (
5  x.  4 )  e.  CC )
82 5nn0 10836 . . . . . . . . . . 11  |-  5  e.  NN0
8315, 82deccl 11014 . . . . . . . . . 10  |- ; 1 5  e.  NN0
8483nn0cni 10828 . . . . . . . . 9  |- ; 1 5  e.  CC
8584a1i 11 . . . . . . . 8  |-  ( n  e.  NN  -> ; 1 5  e.  CC )
8680, 81, 85addsubassd 9970 . . . . . . 7  |-  ( n  e.  NN  ->  (
( ( ( 4 ^ n )  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 5  x.  4 )  - ; 1 5 ) ) )
8779, 86eqtr4d 2501 . . . . . 6  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n
)  x.  4 )  +  ( 5  x.  4 ) )  - ; 1 5 ) )
8857, 62, 873eqtr4rd 2509 . . . . 5  |-  ( n  e.  NN  ->  (
( 4 ^ (
n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n
)  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
8988adantr 465 . . . 4  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
( ( 4 ^ ( n  +  1 ) )  +  5 )  =  ( ( ( ( 4 ^ n )  +  5 )  x.  4 )  -  ( 3  x.  5 ) ) )
9052, 89breqtrrd 4482 . . 3  |-  ( ( n  e.  NN  /\  3  ||  ( ( 4 ^ n )  +  5 ) )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  5 ) )
9190ex 434 . 2  |-  ( n  e.  NN  ->  (
3  ||  ( (
4 ^ n )  +  5 )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  5 ) ) )
923, 6, 9, 12, 34, 91nnind 10574 1  |-  ( N  e.  NN  ->  3  ||  ( ( 4 ^ N )  +  5 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   CCcc 9507   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   NNcn 10556   3c3 10607   4c4 10608   5c5 10609   9c9 10613   NN0cn0 10816   ZZcz 10885  ;cdc 11000   ^cexp 12169    || cdvds 14089
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-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-seq 12111  df-exp 12170  df-dvds 14090
This theorem is referenced by: (None)
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