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Theorem ex-ind-dvds 25899
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 6298 . . . 4  |-  ( k  =  0  ->  (
4 ^ k )  =  ( 4 ^ 0 ) )
21oveq1d 6305 . . 3  |-  ( k  =  0  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ 0 )  +  2 ) )
32breq2d 4414 . 2  |-  ( k  =  0  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ 0 )  +  2 ) ) )
4 oveq2 6298 . . . 4  |-  ( k  =  n  ->  (
4 ^ k )  =  ( 4 ^ n ) )
54oveq1d 6305 . . 3  |-  ( k  =  n  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ n )  +  2 ) )
65breq2d 4414 . 2  |-  ( k  =  n  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ n )  +  2 ) ) )
7 oveq2 6298 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
4 ^ k )  =  ( 4 ^ ( n  +  1 ) ) )
87oveq1d 6305 . . 3  |-  ( k  =  ( n  + 
1 )  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ ( n  + 
1 ) )  +  2 ) )
98breq2d 4414 . 2  |-  ( k  =  ( n  + 
1 )  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  2 ) ) )
10 oveq2 6298 . . . 4  |-  ( k  =  N  ->  (
4 ^ k )  =  ( 4 ^ N ) )
1110oveq1d 6305 . . 3  |-  ( k  =  N  ->  (
( 4 ^ k
)  +  2 )  =  ( ( 4 ^ N )  +  2 ) )
1211breq2d 4414 . 2  |-  ( k  =  N  ->  (
3  ||  ( (
4 ^ k )  +  2 )  <->  3  ||  ( ( 4 ^ N )  +  2 ) ) )
13 3z 10970 . . . 4  |-  3  e.  ZZ
14 iddvds 14316 . . . 4  |-  ( 3  e.  ZZ  ->  3  ||  3 )
1513, 14ax-mp 5 . . 3  |-  3  ||  3
16 4nn0 10888 . . . . . 6  |-  4  e.  NN0
1716numexp0 15048 . . . . 5  |-  ( 4 ^ 0 )  =  1
1817oveq1i 6300 . . . 4  |-  ( ( 4 ^ 0 )  +  2 )  =  ( 1  +  2 )
19 1p2e3 10734 . . . 4  |-  ( 1  +  2 )  =  3
2018, 19eqtri 2473 . . 3  |-  ( ( 4 ^ 0 )  +  2 )  =  3
2115, 20breqtrri 4428 . 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 12438 . . . . . . . . 9  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e. 
NN0 )
2625nn0zd 11038 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e.  ZZ )
2726adantr 467 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  ZZ )
28 2z 10969 . . . . . . . 8  |-  2  e.  ZZ
2928a1i 11 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
2  e.  ZZ )
3027, 29zaddcld 11044 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ n )  +  2 )  e.  ZZ )
31 4z 10971 . . . . . . 7  |-  4  e.  ZZ
3231a1i 11 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
4  e.  ZZ )
33 simpr 463 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
4 ^ n )  +  2 ) )
3422, 30, 32, 33dvdsmultr1d 14339 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
( 4 ^ n
)  +  2 )  x.  4 ) )
35 dvdsmul1 14324 . . . . . . 7  |-  ( ( 3  e.  ZZ  /\  2  e.  ZZ )  ->  3  ||  ( 3  x.  2 ) )
3613, 28, 35mp2an 678 . . . . . 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 459 . . . . . . . . 9  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  ->  n  e.  NN0 )
4038, 39nn0expcld 12438 . . . . . . . 8  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  NN0 )
4140nn0zd 11038 . . . . . . 7  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 4 ^ n
)  e.  ZZ )
4241, 29zaddcld 11044 . . . . . 6  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ n )  +  2 )  e.  ZZ )
4342, 32zmulcld 11046 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( ( 4 ^ n )  +  2 )  x.  4 )  e.  ZZ )
4422, 29zmulcld 11046 . . . . 5  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( 3  x.  2 )  e.  ZZ )
4522, 34, 37, 43, 44dvds2subd 14336 . . . 4  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) ) )
4625nn0cnd 10927 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ n )  e.  CC )
47 2cnd 10682 . . . . . . . 8  |-  ( n  e.  NN0  ->  2  e.  CC )
48 4cn 10687 . . . . . . . . 9  |-  4  e.  CC
4948a1i 11 . . . . . . . 8  |-  ( n  e.  NN0  ->  4  e.  CC )
5046, 47, 49adddird 9668 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( ( 4 ^ n
)  +  2 )  x.  4 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) ) )
5150oveq1d 6305 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 2  x.  3 ) )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) )  -  (
2  x.  3 ) ) )
52 3cn 10684 . . . . . . . . 9  |-  3  e.  CC
53 2cn 10680 . . . . . . . . 9  |-  2  e.  CC
5452, 53mulcomi 9649 . . . . . . . 8  |-  ( 3  x.  2 )  =  ( 2  x.  3 )
5554a1i 11 . . . . . . 7  |-  ( n  e.  NN0  ->  ( 3  x.  2 )  =  ( 2  x.  3 ) )
5655oveq2d 6306 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  (
2  x.  3 ) ) )
5749, 24expp1d 12417 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 4 ^ ( n  + 
1 ) )  =  ( ( 4 ^ n )  x.  4 ) )
58 ax-1cn 9597 . . . . . . . . . . . . . . 15  |-  1  e.  CC
59 3p1e4 10735 . . . . . . . . . . . . . . 15  |-  ( 3  +  1 )  =  4
6052, 58, 59addcomli 9825 . . . . . . . . . . . . . 14  |-  ( 1  +  3 )  =  4
6160eqcomi 2460 . . . . . . . . . . . . 13  |-  4  =  ( 1  +  3 )
6261oveq1i 6300 . . . . . . . . . . . 12  |-  ( 4  -  3 )  =  ( ( 1  +  3 )  -  3 )
6358, 52pncan3oi 9891 . . . . . . . . . . . 12  |-  ( ( 1  +  3 )  -  3 )  =  1
6462, 63eqtri 2473 . . . . . . . . . . 11  |-  ( 4  -  3 )  =  1
6564oveq2i 6301 . . . . . . . . . 10  |-  ( 2  x.  ( 4  -  3 ) )  =  ( 2  x.  1 )
6653, 48, 52subdii 10067 . . . . . . . . . 10  |-  ( 2  x.  ( 4  -  3 ) )  =  ( ( 2  x.  4 )  -  (
2  x.  3 ) )
67 2t1e2 10758 . . . . . . . . . 10  |-  ( 2  x.  1 )  =  2
6865, 66, 673eqtr3ri 2482 . . . . . . . . 9  |-  2  =  ( ( 2  x.  4 )  -  ( 2  x.  3 ) )
6968a1i 11 . . . . . . . 8  |-  ( n  e.  NN0  ->  2  =  ( ( 2  x.  4 )  -  (
2  x.  3 ) ) )
7057, 69oveq12d 6308 . . . . . . 7  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( 4 ^ n )  x.  4 )  +  ( ( 2  x.  4 )  -  ( 2  x.  3 ) ) ) )
7146, 49mulcld 9663 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( ( 4 ^ n )  x.  4 )  e.  CC )
7247, 49mulcld 9663 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 2  x.  4 )  e.  CC )
7352a1i 11 . . . . . . . . 9  |-  ( n  e.  NN0  ->  3  e.  CC )
7447, 73mulcld 9663 . . . . . . . 8  |-  ( n  e.  NN0  ->  ( 2  x.  3 )  e.  CC )
7571, 72, 74addsubassd 10006 . . . . . . 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 2488 . . . . . 6  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  x.  4 )  +  ( 2  x.  4 ) )  -  (
2  x.  3 ) ) )
7751, 56, 763eqtr4rd 2496 . . . . 5  |-  ( n  e.  NN0  ->  ( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  (
3  x.  2 ) ) )
7877adantr 467 . . . 4  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
( ( 4 ^ ( n  +  1 ) )  +  2 )  =  ( ( ( ( 4 ^ n )  +  2 )  x.  4 )  -  ( 3  x.  2 ) ) )
7945, 78breqtrrd 4429 . . 3  |-  ( ( n  e.  NN0  /\  3  ||  ( ( 4 ^ n )  +  2 ) )  -> 
3  ||  ( (
4 ^ ( n  +  1 ) )  +  2 ) )
8079ex 436 . 2  |-  ( n  e.  NN0  ->  ( 3 
||  ( ( 4 ^ n )  +  2 )  ->  3  ||  ( ( 4 ^ ( n  +  1 ) )  +  2 ) ) )
813, 6, 9, 12, 21, 80nn0ind 11030 1  |-  ( N  e.  NN0  ->  3  ||  ( ( 4 ^ N )  +  2 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1444    e. wcel 1887   class class class wbr 4402  (class class class)co 6290   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    x. cmul 9544    - cmin 9860   2c2 10659   3c3 10660   4c4 10661   NN0cn0 10869   ZZcz 10937   ^cexp 12272    || cdvds 14305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-n0 10870  df-z 10938  df-uz 11160  df-seq 12214  df-exp 12273  df-dvds 14306
This theorem is referenced by: (None)
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