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Theorem ex-ind-dvds 25388
Description: Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
Assertion
Ref Expression
ex-ind-dvds  |-  ( N  e.  NN0  ->  3  ||  ( ( 4 ^ N )  +  2 ) )

Proof of Theorem ex-ind-dvds
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6304 . . . 4  |-  ( k  =  0  ->  (
4 ^ k )  =  ( 4 ^ 0 ) )
21oveq1d 6311 . . 3  |-  ( k  =  0  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ 0 )  +  2 ) )
32breq2d 4468 . 2  |-  ( k  =  0  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ 0 )  +  2 ) ) )
4 oveq2 6304 . . . 4  |-  ( k  =  n  ->  (
4 ^ k )  =  ( 4 ^ n ) )
54oveq1d 6311 . . 3  |-  ( k  =  n  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ n )  +  2 ) )
65breq2d 4468 . 2  |-  ( k  =  n  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ n )  +  2 ) ) )
7 oveq2 6304 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
4 ^ k )  =  ( 4 ^ ( n  +  1 ) ) )
87oveq1d 6311 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ ( n  + 
1 ) )  +  2 ) )
98breq2d 4468 . 2  |-  ( k  =  ( n  + 
1 )  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  2 ) ) )
10 oveq2 6304 . . . 4  |-  ( k  =  N  ->  (
4 ^ k )  =  ( 4 ^ N ) )
1110oveq1d 6311 . . 3  |-  ( k  =  N  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ N )  +  2 ) )
1211breq2d 4468 . 2  |-  ( k  =  N  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ N )  +  2 ) ) )
13 3z 10918 . . . 4  |-  3  e.  ZZ
14 iddvds 14100 . . . 4  |-  ( 3  e.  ZZ  ->  3  ||  3 )
1513, 14ax-mp 5 . . 3  |-  3  ||  3
16 4nn0 10835 . . . . . 6  |-  4  e.  NN0
1716numexp0 14665 . . . . 5  |-  ( 4 ^ 0 )  =  1
1817oveq1i 6306 . . . 4  |-  ( ( 4 ^ 0 )  +  2 )  =  ( 1  +  2 )
19 1p2e3 10681 . . . 4  |-  ( 1  +  2 )  =  3
2018, 19eqtri 2486 . . 3  |-  ( ( 4 ^ 0 )  +  2 )  =  3
2115, 20breqtrri 4481 . 2  |-  3  ||  ( ( 4 ^ 0 )  +  2 )
2213a1i 11 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  e.  ZZ )
2316a1i 11 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  4  e. 
NN0 )
24 id 22 . . . . . . . . . 10  |-  ( n  e.  NN0  ->  n  e. 
NN0 )
2523, 24nn0expcld 12335 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e. 
NN0 )
2625nn0zd 10988 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e.  ZZ )
2726adantr 465 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  ZZ )
28 2z 10917 . . . . . . . 8  |-  2  e.  ZZ
2928a1i 11 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
2  e.  ZZ )
3027, 29zaddcld 10994 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ n )  +  2 )  e.  ZZ )
31 4z 10919 . . . . . . 7  |-  4  e.  ZZ
3231a1i 11 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
4  e.  ZZ )
33 simpr 461 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
4 ^ n )  +  2 ) )
3422, 30, 32, 33dvdsmultr1d 14123 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
( 4 ^ n
)  +  2 )  x.  4 ) )
35 dvdsmul1 14108 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  3  ||  ( 3  x.  2 ) )
3613, 28, 35mp2an 672 . . . . . 6  |-  3  ||  ( 3  x.  2 )
3736a1i 11 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( 3  x.  2 ) )
3816a1i 11 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
4  e.  NN0 )
39 simpl 457 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  ->  n  e.  NN0 )
4038, 39nn0expcld 12335 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  NN0 )
4140nn0zd 10988 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  ZZ )
4241, 29zaddcld 10994 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ n )  +  2 )  e.  ZZ )
4342, 32zmulcld 10996 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( ( 4 ^ n )  +  2 )  x.  4 )  e.  ZZ )
4422, 29zmulcld 10996 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 3  x.  2 )  e.  ZZ )
4522, 34, 37, 43, 44dvds2subd 14120 . . . 4  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) ) )
4625nn0cnd 10875 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e.  CC )
47 2cnd 10629 . . . . . . . 8  |-  ( n  e.  NN0  ->  2  e.  CC )
48 4cn 10634 . . . . . . . . 9  |-  4  e.  CC
4948a1i 11 . . . . . . . 8  |-  ( n  e.  NN0  ->  4  e.  CC )
5046, 47, 49adddird 9638 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( ( 4 ^ n
)  +  2 )  x.  4 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) ) )
5150oveq1d 6311 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 2  x.  3 ) )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) )  -  (
2  x.  3 ) ) )
52 3cn 10631 . . . . . . . . 9  |-  3  e.  CC
53 2cn 10627 . . . . . . . . 9  |-  2  e.  CC
5452, 53mulcomi 9619 . . . . . . . 8  |-  ( 3  x.  2 )  =  ( 2  x.  3 )
5554a1i 11 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 3  x.  2 )  =  ( 2  x.  3 ) )
5655oveq2d 6312 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  (
2  x.  3 ) ) )
5749, 24expp1d 12314 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ ( n  + 
1 ) )  =  ( ( 4 ^ n )  x.  4 ) )
58 ax-1cn 9567 . . . . . . . . . . . . . . 15  |-  1  e.  CC
59 3p1e4 10682 . . . . . . . . . . . . . . 15  |-  ( 3  +  1 )  =  4
6052, 58, 59addcomli 9789 . . . . . . . . . . . . . 14  |-  ( 1  +  3 )  =  4
6160eqcomi 2470 . . . . . . . . . . . . 13  |-  4  =  ( 1  +  3 )
6261oveq1i 6306 . . . . . . . . . . . 12  |-  ( 4  -  3 )  =  ( ( 1  +  3 )  -  3 )
6358, 52pncan3oi 9855 . . . . . . . . . . . 12  |-  ( ( 1  +  3 )  -  3 )  =  1
6462, 63eqtri 2486 . . . . . . . . . . 11  |-  ( 4  -  3 )  =  1
6564oveq2i 6307 . . . . . . . . . 10  |-  ( 2  x.  ( 4  -  3 ) )  =  ( 2  x.  1 )
6653, 48, 52subdii 10026 . . . . . . . . . 10  |-  ( 2  x.  ( 4  -  3 ) )  =  ( ( 2  x.  4 )  -  (
2  x.  3 ) )
67 2t1e2 10705 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
6865, 66, 673eqtr3ri 2495 . . . . . . . . 9  |-  2  =  ( ( 2  x.  4 )  -  ( 2  x.  3 ) )
6968a1i 11 . . . . . . . 8  |-  ( n  e.  NN0  ->  2  =  ( ( 2  x.  4 )  -  (
2  x.  3 ) ) )
7057, 69oveq12d 6314 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 2  x.  4 )  -  ( 2  x.  3 ) ) ) )
7146, 49mulcld 9633 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 4 ^ n )  x.  4 )  e.  CC )
7247, 49mulcld 9633 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 2  x.  4 )  e.  CC )
7352a1i 11 . . . . . . . . 9  |-  ( n  e.  NN0  ->  3  e.  CC )
7447, 73mulcld 9633 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 2  x.  3 )  e.  CC )
7571, 72, 74addsubassd 9970 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) )  -  ( 2  x.  3 ) )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 2  x.  4 )  -  ( 2  x.  3 ) ) ) )
7670, 75eqtr4d 2501 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) )  -  (
2  x.  3 ) ) )
7751, 56, 763eqtr4rd 2509 . . . . 5  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  (
3  x.  2 ) ) )
7877adantr 465 . . . 4  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) ) )
7945, 78breqtrrd 4482 . . 3  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  2 ) )
8079ex 434 . 2  |-  ( n  e.  NN0  ->  ( 3 
||  ( ( 4 ^ n )  +  2 )  ->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  2 ) ) )
813, 6, 9, 12, 21, 80nn0ind 10980 1  |-  ( N  e.  NN0  ->  3  ||  ( ( 4 ^ N )  +  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514    - cmin 9824   2c2 10606   3c3 10607   4c4 10608   NN0cn0 10816   ZZcz 10885   ^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-n0 10817  df-z 10886  df-uz 11107  df-seq 12111  df-exp 12170  df-dvds 14090
This theorem is referenced by: (None)
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