Theorem ex-ind-dvds 27318

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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( k = 0 -> ( 4 ^ k ) = ( 4 ^ 0 ) )
2	1	oveq1d 6665	. . 3 |- ( k = 0 -> ( ( 4 ^ k ) + 2 ) = ( ( 4 ^ 0 ) + 2 ) )
3	2	breq2d 4665	. 2 |- ( k = 0 -> ( 3 || ( ( 4 ^ k ) + 2 ) <-> 3 || ( ( 4 ^ 0 ) + 2 ) ) )
4		oveq2 6658	. . . 4 |- ( k = n -> ( 4 ^ k ) = ( 4 ^ n ) )
5	4	oveq1d 6665	. . 3 |- ( k = n -> ( ( 4 ^ k ) + 2 ) = ( ( 4 ^ n ) + 2 ) )
6	5	breq2d 4665	. 2 |- ( k = n -> ( 3 || ( ( 4 ^ k ) + 2 ) <-> 3 || ( ( 4 ^ n ) + 2 ) ) )
7		oveq2 6658	. . . 4 |- ( k = ( n + 1 ) -> ( 4 ^ k ) = ( 4 ^ ( n + 1 ) ) )
8	7	oveq1d 6665	. . 3 |- ( k = ( n + 1 ) -> ( ( 4 ^ k ) + 2 ) = ( ( 4 ^ ( n + 1 ) ) + 2 ) )
9	8	breq2d 4665	. 2 |- ( k = ( n + 1 ) -> ( 3 || ( ( 4 ^ k ) + 2 ) <-> 3 || ( ( 4 ^ ( n + 1 ) ) + 2 ) ) )
10		oveq2 6658	. . . 4 |- ( k = N -> ( 4 ^ k ) = ( 4 ^ N ) )
11	10	oveq1d 6665	. . 3 |- ( k = N -> ( ( 4 ^ k ) + 2 ) = ( ( 4 ^ N ) + 2 ) )
12	11	breq2d 4665	. 2 |- ( k = N -> ( 3 || ( ( 4 ^ k ) + 2 ) <-> 3 || ( ( 4 ^ N ) + 2 ) ) )
13		3z 11410	. . . 4 |- 3 e. ZZ
14		iddvds 14995	. . . 4 |- ( 3 e. ZZ -> 3 || 3 )
15	13, 14	ax-mp 5	. . 3 |- 3 || 3
16		4nn0 11311	. . . . . 6 |- 4 e. NN0
17	16	numexp0 15780	. . . . 5 |- ( 4 ^ 0 ) = 1
18	17	oveq1i 6660	. . . 4 |- ( ( 4 ^ 0 ) + 2 ) = ( 1 + 2 )
19		1p2e3 11152	. . . 4 |- ( 1 + 2 ) = 3
20	18, 19	eqtri 2644	. . 3 |- ( ( 4 ^ 0 ) + 2 ) = 3
21	15, 20	breqtrri 4680	. 2 |- 3 || ( ( 4 ^ 0 ) + 2 )
22	13	a1i 11	. . . . 5 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 e. ZZ )
23	16	a1i 11	. . . . . . . . . 10 |- ( n e. NN0 -> 4 e. NN0 )
24		id 22	. . . . . . . . . 10 |- ( n e. NN0 -> n e. NN0 )
25	23, 24	nn0expcld 13031	. . . . . . . . 9 |- ( n e. NN0 -> ( 4 ^ n ) e. NN0 )
26	25	nn0zd 11480	. . . . . . . 8 |- ( n e. NN0 -> ( 4 ^ n ) e. ZZ )
27	26	adantr 481	. . . . . . 7 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( 4 ^ n ) e. ZZ )
28		2z 11409	. . . . . . . 8 |- 2 e. ZZ
29	28	a1i 11	. . . . . . 7 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 2 e. ZZ )
30	27, 29	zaddcld 11486	. . . . . 6 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( ( 4 ^ n ) + 2 ) e. ZZ )
31		4z 11411	. . . . . . 7 |- 4 e. ZZ
32	31	a1i 11	. . . . . 6 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 4 e. ZZ )
33		simpr 477	. . . . . 6 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 || ( ( 4 ^ n ) + 2 ) )
34	22, 30, 32, 33	dvdsmultr1d 15020	. . . . 5 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 || ( ( ( 4 ^ n ) + 2 ) x. 4 ) )
35		dvdsmul1 15003	. . . . . . 7 |- ( ( 3 e. ZZ /\ 2 e. ZZ ) -> 3 || ( 3 x. 2 ) )
36	13, 28, 35	mp2an 708	. . . . . 6 |- 3 || ( 3 x. 2 )
37	36	a1i 11	. . . . 5 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 || ( 3 x. 2 ) )
38	16	a1i 11	. . . . . . . . 9 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 4 e. NN0 )
39		simpl 473	. . . . . . . . 9 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> n e. NN0 )
40	38, 39	nn0expcld 13031	. . . . . . . 8 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( 4 ^ n ) e. NN0 )
41	40	nn0zd 11480	. . . . . . 7 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( 4 ^ n ) e. ZZ )
42	41, 29	zaddcld 11486	. . . . . 6 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( ( 4 ^ n ) + 2 ) e. ZZ )
43	42, 32	zmulcld 11488	. . . . 5 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( ( ( 4 ^ n ) + 2 ) x. 4 ) e. ZZ )
44	22, 29	zmulcld 11488	. . . . 5 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( 3 x. 2 ) e. ZZ )
45	22, 34, 37, 43, 44	dvds2subd 15017	. . . 4 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 || ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 3 x. 2 ) ) )
46	25	nn0cnd 11353	. . . . . . . 8 |- ( n e. NN0 -> ( 4 ^ n ) e. CC )
47		2cnd 11093	. . . . . . . 8 |- ( n e. NN0 -> 2 e. CC )
48		4cn 11098	. . . . . . . . 9 |- 4 e. CC
49	48	a1i 11	. . . . . . . 8 |- ( n e. NN0 -> 4 e. CC )
50	46, 47, 49	adddird 10065	. . . . . . 7 |- ( n e. NN0 -> ( ( ( 4 ^ n ) + 2 ) x. 4 ) = ( ( ( 4 ^ n ) x. 4 ) + ( 2 x. 4 ) ) )
51	50	oveq1d 6665	. . . . . 6 |- ( n e. NN0 -> ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 2 x. 3 ) ) = ( ( ( ( 4 ^ n ) x. 4 ) + ( 2 x. 4 ) ) - ( 2 x. 3 ) ) )
52		3cn 11095	. . . . . . . . 9 |- 3 e. CC
53		2cn 11091	. . . . . . . . 9 |- 2 e. CC
54	52, 53	mulcomi 10046	. . . . . . . 8 |- ( 3 x. 2 ) = ( 2 x. 3 )
55	54	a1i 11	. . . . . . 7 |- ( n e. NN0 -> ( 3 x. 2 ) = ( 2 x. 3 ) )
56	55	oveq2d 6666	. . . . . 6 |- ( n e. NN0 -> ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 3 x. 2 ) ) = ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 2 x. 3 ) ) )
57	49, 24	expp1d 13009	. . . . . . . 8 |- ( n e. NN0 -> ( 4 ^ ( n + 1 ) ) = ( ( 4 ^ n ) x. 4 ) )
58		ax-1cn 9994	. . . . . . . . . . . . . . 15 |- 1 e. CC
59		3p1e4 11153	. . . . . . . . . . . . . . 15 |- ( 3 + 1 ) = 4
60	52, 58, 59	addcomli 10228	. . . . . . . . . . . . . 14 |- ( 1 + 3 ) = 4
61	60	eqcomi 2631	. . . . . . . . . . . . 13 |- 4 = ( 1 + 3 )
62	61	oveq1i 6660	. . . . . . . . . . . 12 |- ( 4 - 3 ) = ( ( 1 + 3 ) - 3 )
63	58, 52	pncan3oi 10297	. . . . . . . . . . . 12 |- ( ( 1 + 3 ) - 3 ) = 1
64	62, 63	eqtri 2644	. . . . . . . . . . 11 |- ( 4 - 3 ) = 1
65	64	oveq2i 6661	. . . . . . . . . 10 |- ( 2 x. ( 4 - 3 ) ) = ( 2 x. 1 )
66	53, 48, 52	subdii 10479	. . . . . . . . . 10 |- ( 2 x. ( 4 - 3 ) ) = ( ( 2 x. 4 ) - ( 2 x. 3 ) )
67		2t1e2 11176	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
68	65, 66, 67	3eqtr3ri 2653	. . . . . . . . 9 |- 2 = ( ( 2 x. 4 ) - ( 2 x. 3 ) )
69	68	a1i 11	. . . . . . . 8 |- ( n e. NN0 -> 2 = ( ( 2 x. 4 ) - ( 2 x. 3 ) ) )
70	57, 69	oveq12d 6668	. . . . . . 7 |- ( n e. NN0 -> ( ( 4 ^ ( n + 1 ) ) + 2 ) = ( ( ( 4 ^ n ) x. 4 ) + ( ( 2 x. 4 ) - ( 2 x. 3 ) ) ) )
71	46, 49	mulcld 10060	. . . . . . . 8 |- ( n e. NN0 -> ( ( 4 ^ n ) x. 4 ) e. CC )
72	47, 49	mulcld 10060	. . . . . . . 8 |- ( n e. NN0 -> ( 2 x. 4 ) e. CC )
73	52	a1i 11	. . . . . . . . 9 |- ( n e. NN0 -> 3 e. CC )
74	47, 73	mulcld 10060	. . . . . . . 8 |- ( n e. NN0 -> ( 2 x. 3 ) e. CC )
75	71, 72, 74	addsubassd 10412	. . . . . . 7 |- ( n e. NN0 -> ( ( ( ( 4 ^ n ) x. 4 ) + ( 2 x. 4 ) ) - ( 2 x. 3 ) ) = ( ( ( 4 ^ n ) x. 4 ) + ( ( 2 x. 4 ) - ( 2 x. 3 ) ) ) )
76	70, 75	eqtr4d 2659	. . . . . 6 |- ( n e. NN0 -> ( ( 4 ^ ( n + 1 ) ) + 2 ) = ( ( ( ( 4 ^ n ) x. 4 ) + ( 2 x. 4 ) ) - ( 2 x. 3 ) ) )
77	51, 56, 76	3eqtr4rd 2667	. . . . 5 |- ( n e. NN0 -> ( ( 4 ^ ( n + 1 ) ) + 2 ) = ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 3 x. 2 ) ) )
78	77	adantr 481	. . . 4 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> ( ( 4 ^ ( n + 1 ) ) + 2 ) = ( ( ( ( 4 ^ n ) + 2 ) x. 4 ) - ( 3 x. 2 ) ) )
79	45, 78	breqtrrd 4681	. . 3 |- ( ( n e. NN0 /\ 3 || ( ( 4 ^ n ) + 2 ) ) -> 3 || ( ( 4 ^ ( n + 1 ) ) + 2 ) )
80	79	ex 450	. 2 |- ( n e. NN0 -> ( 3 || ( ( 4 ^ n ) + 2 ) -> 3 || ( ( 4 ^ ( n + 1 ) ) + 2 ) ) )
81	3, 6, 9, 12, 21, 80	nn0ind 11472	1 |- ( N e. NN0 -> 3 || ( ( 4 ^ N ) + 2 ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   2c2 11070   3c3 11071   4c4 11072   NN0cn0 11292   ZZcz 11377   ^cexp 12860   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861  df-dvds 14984

This theorem is referenced by: (None)