Theorem inductionexd 38453

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.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . 4 |- ( k = 1 -> ( 4 ^ k ) = ( 4 ^ 1 ) )
2	1	oveq1d 6665	. . 3 |- ( k = 1 -> ( ( 4 ^ k ) + 5 ) = ( ( 4 ^ 1 ) + 5 ) )
3	2	breq2d 4665	. 2 |- ( k = 1 -> ( 3 || ( ( 4 ^ k ) + 5 ) <-> 3 || ( ( 4 ^ 1 ) + 5 ) ) )
4		oveq2 6658	. . . 4 |- ( k = n -> ( 4 ^ k ) = ( 4 ^ n ) )
5	4	oveq1d 6665	. . 3 |- ( k = n -> ( ( 4 ^ k ) + 5 ) = ( ( 4 ^ n ) + 5 ) )
6	5	breq2d 4665	. 2 |- ( k = n -> ( 3 || ( ( 4 ^ k ) + 5 ) <-> 3 || ( ( 4 ^ n ) + 5 ) ) )
7		oveq2 6658	. . . 4 |- ( k = ( n + 1 ) -> ( 4 ^ k ) = ( 4 ^ ( n + 1 ) ) )
8	7	oveq1d 6665	. . 3 |- ( k = ( n + 1 ) -> ( ( 4 ^ k ) + 5 ) = ( ( 4 ^ ( n + 1 ) ) + 5 ) )
9	8	breq2d 4665	. 2 |- ( k = ( n + 1 ) -> ( 3 || ( ( 4 ^ k ) + 5 ) <-> 3 || ( ( 4 ^ ( n + 1 ) ) + 5 ) ) )
10		oveq2 6658	. . . 4 |- ( k = N -> ( 4 ^ k ) = ( 4 ^ N ) )
11	10	oveq1d 6665	. . 3 |- ( k = N -> ( ( 4 ^ k ) + 5 ) = ( ( 4 ^ N ) + 5 ) )
12	11	breq2d 4665	. 2 |- ( k = N -> ( 3 || ( ( 4 ^ k ) + 5 ) <-> 3 || ( ( 4 ^ N ) + 5 ) ) )
13		3z 11410	. . . 4 |- 3 e. ZZ
14		4z 11411	. . . . . 6 |- 4 e. ZZ
15		1nn0 11308	. . . . . 6 |- 1 e. NN0
16		zexpcl 12875	. . . . . 6 |- ( ( 4 e. ZZ /\ 1 e. NN0 ) -> ( 4 ^ 1 ) e. ZZ )
17	14, 15, 16	mp2an 708	. . . . 5 |- ( 4 ^ 1 ) e. ZZ
18		5nn 11188	. . . . . 6 |- 5 e. NN
19	18	nnzi 11401	. . . . 5 |- 5 e. ZZ
20		zaddcl 11417	. . . . 5 |- ( ( ( 4 ^ 1 ) e. ZZ /\ 5 e. ZZ ) -> ( ( 4 ^ 1 ) + 5 ) e. ZZ )
21	17, 19, 20	mp2an 708	. . . 4 |- ( ( 4 ^ 1 ) + 5 ) e. ZZ
22	13, 13, 21	3pm3.2i 1239	. . 3 |- ( 3 e. ZZ /\ 3 e. ZZ /\ ( ( 4 ^ 1 ) + 5 ) e. ZZ )
23		3t3e9 11180	. . . 4 |- ( 3 x. 3 ) = 9
24		4nn0 11311	. . . . . . 7 |- 4 e. NN0
25	24	numexp1 15781	. . . . . 6 |- ( 4 ^ 1 ) = 4
26	25	oveq1i 6660	. . . . 5 |- ( ( 4 ^ 1 ) + 5 ) = ( 4 + 5 )
27		5cn 11100	. . . . . 6 |- 5 e. CC
28		4cn 11098	. . . . . 6 |- 4 e. CC
29		5p4e9 11167	. . . . . 6 |- ( 5 + 4 ) = 9
30	27, 28, 29	addcomli 10228	. . . . 5 |- ( 4 + 5 ) = 9
31	26, 30	eqtri 2644	. . . 4 |- ( ( 4 ^ 1 ) + 5 ) = 9
32	23, 31	eqtr4i 2647	. . 3 |- ( 3 x. 3 ) = ( ( 4 ^ 1 ) + 5 )
33		dvds0lem 14992	. . 3 |- ( ( ( 3 e. ZZ /\ 3 e. ZZ /\ ( ( 4 ^ 1 ) + 5 ) e. ZZ ) /\ ( 3 x. 3 ) = ( ( 4 ^ 1 ) + 5 ) ) -> 3 || ( ( 4 ^ 1 ) + 5 ) )
34	22, 32, 33	mp2an 708	. 2 |- 3 || ( ( 4 ^ 1 ) + 5 )
35	13	a1i 11	. . . . 5 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 e. ZZ )
36		4nn 11187	. . . . . . . . . . 11 |- 4 e. NN
37	36	a1i 11	. . . . . . . . . 10 |- ( n e. NN -> 4 e. NN )
38		nnnn0 11299	. . . . . . . . . 10 |- ( n e. NN -> n e. NN0 )
39	37, 38	nnexpcld 13030	. . . . . . . . 9 |- ( n e. NN -> ( 4 ^ n ) e. NN )
40	39	nnzd 11481	. . . . . . . 8 |- ( n e. NN -> ( 4 ^ n ) e. ZZ )
41	40	adantr 481	. . . . . . 7 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> ( 4 ^ n ) e. ZZ )
42	19	a1i 11	. . . . . . 7 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 5 e. ZZ )
43	41, 42	zaddcld 11486	. . . . . 6 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> ( ( 4 ^ n ) + 5 ) e. ZZ )
44	14	a1i 11	. . . . . 6 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 4 e. ZZ )
45		simpr 477	. . . . . 6 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 || ( ( 4 ^ n ) + 5 ) )
46	35, 43, 44, 45	dvdsmultr1d 15020	. . . . 5 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 || ( ( ( 4 ^ n ) + 5 ) x. 4 ) )
47		dvdsmul1 15003	. . . . . . 7 |- ( ( 3 e. ZZ /\ 5 e. ZZ ) -> 3 || ( 3 x. 5 ) )
48	13, 19, 47	mp2an 708	. . . . . 6 |- 3 || ( 3 x. 5 )
49	48	a1i 11	. . . . 5 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 || ( 3 x. 5 ) )
50	43, 44	zmulcld 11488	. . . . 5 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> ( ( ( 4 ^ n ) + 5 ) x. 4 ) e. ZZ )
51	35, 42	zmulcld 11488	. . . . 5 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> ( 3 x. 5 ) e. ZZ )
52	35, 46, 49, 50, 51	dvds2subd 15017	. . . 4 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 || ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ( 3 x. 5 ) ) )
53	39	nncnd 11036	. . . . . . . 8 |- ( n e. NN -> ( 4 ^ n ) e. CC )
54	27	a1i 11	. . . . . . . 8 |- ( n e. NN -> 5 e. CC )
55	28	a1i 11	. . . . . . . 8 |- ( n e. NN -> 4 e. CC )
56	53, 54, 55	adddird 10065	. . . . . . 7 |- ( n e. NN -> ( ( ( 4 ^ n ) + 5 ) x. 4 ) = ( ( ( 4 ^ n ) x. 4 ) + ( 5 x. 4 ) ) )
57	56	oveq1d 6665	. . . . . 6 |- ( n e. NN -> ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ; 1 5 ) = ( ( ( ( 4 ^ n ) x. 4 ) + ( 5 x. 4 ) ) - ; 1 5 ) )
58		3cn 11095	. . . . . . . . 9 |- 3 e. CC
59		5t3e15 11635	. . . . . . . . 9 |- ( 5 x. 3 ) = ; 1 5
60	27, 58, 59	mulcomli 10047	. . . . . . . 8 |- ( 3 x. 5 ) = ; 1 5
61	60	a1i 11	. . . . . . 7 |- ( n e. NN -> ( 3 x. 5 ) = ; 1 5 )
62	61	oveq2d 6666	. . . . . 6 |- ( n e. NN -> ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ( 3 x. 5 ) ) = ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ; 1 5 ) )
63	55, 38	expp1d 13009	. . . . . . . 8 |- ( n e. NN -> ( 4 ^ ( n + 1 ) ) = ( ( 4 ^ n ) x. 4 ) )
64		ax-1cn 9994	. . . . . . . . . . . . . . . 16 |- 1 e. CC
65		3p1e4 11153	. . . . . . . . . . . . . . . 16 |- ( 3 + 1 ) = 4
66	58, 64, 65	addcomli 10228	. . . . . . . . . . . . . . 15 |- ( 1 + 3 ) = 4
67	66	eqcomi 2631	. . . . . . . . . . . . . 14 |- 4 = ( 1 + 3 )
68	67	oveq1i 6660	. . . . . . . . . . . . 13 |- ( 4 - 3 ) = ( ( 1 + 3 ) - 3 )
69	64, 58	pncan3oi 10297	. . . . . . . . . . . . 13 |- ( ( 1 + 3 ) - 3 ) = 1
70	68, 69	eqtri 2644	. . . . . . . . . . . 12 |- ( 4 - 3 ) = 1
71	70	oveq2i 6661	. . . . . . . . . . 11 |- ( 5 x. ( 4 - 3 ) ) = ( 5 x. 1 )
72	27, 28, 58	subdii 10479	. . . . . . . . . . 11 |- ( 5 x. ( 4 - 3 ) ) = ( ( 5 x. 4 ) - ( 5 x. 3 ) )
73	27	mulid1i 10042	. . . . . . . . . . 11 |- ( 5 x. 1 ) = 5
74	71, 72, 73	3eqtr3ri 2653	. . . . . . . . . 10 |- 5 = ( ( 5 x. 4 ) - ( 5 x. 3 ) )
75	59	eqcomi 2631	. . . . . . . . . . 11 |- ; 1 5 = ( 5 x. 3 )
76	75	oveq2i 6661	. . . . . . . . . 10 |- ( ( 5 x. 4 ) - ; 1 5 ) = ( ( 5 x. 4 ) - ( 5 x. 3 ) )
77	74, 76	eqtr4i 2647	. . . . . . . . 9 |- 5 = ( ( 5 x. 4 ) - ; 1 5 )
78	77	a1i 11	. . . . . . . 8 |- ( n e. NN -> 5 = ( ( 5 x. 4 ) - ; 1 5 ) )
79	63, 78	oveq12d 6668	. . . . . . 7 |- ( n e. NN -> ( ( 4 ^ ( n + 1 ) ) + 5 ) = ( ( ( 4 ^ n ) x. 4 ) + ( ( 5 x. 4 ) - ; 1 5 ) ) )
80	53, 55	mulcld 10060	. . . . . . . 8 |- ( n e. NN -> ( ( 4 ^ n ) x. 4 ) e. CC )
81	54, 55	mulcld 10060	. . . . . . . 8 |- ( n e. NN -> ( 5 x. 4 ) e. CC )
82		5nn0 11312	. . . . . . . . . . 11 |- 5 e. NN0
83	15, 82	deccl 11512	. . . . . . . . . 10 |- ; 1 5 e. NN0
84	83	nn0cni 11304	. . . . . . . . 9 |- ; 1 5 e. CC
85	84	a1i 11	. . . . . . . 8 |- ( n e. NN -> ; 1 5 e. CC )
86	80, 81, 85	addsubassd 10412	. . . . . . 7 |- ( n e. NN -> ( ( ( ( 4 ^ n ) x. 4 ) + ( 5 x. 4 ) ) - ; 1 5 ) = ( ( ( 4 ^ n ) x. 4 ) + ( ( 5 x. 4 ) - ; 1 5 ) ) )
87	79, 86	eqtr4d 2659	. . . . . 6 |- ( n e. NN -> ( ( 4 ^ ( n + 1 ) ) + 5 ) = ( ( ( ( 4 ^ n ) x. 4 ) + ( 5 x. 4 ) ) - ; 1 5 ) )
88	57, 62, 87	3eqtr4rd 2667	. . . . 5 |- ( n e. NN -> ( ( 4 ^ ( n + 1 ) ) + 5 ) = ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ( 3 x. 5 ) ) )
89	88	adantr 481	. . . 4 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> ( ( 4 ^ ( n + 1 ) ) + 5 ) = ( ( ( ( 4 ^ n ) + 5 ) x. 4 ) - ( 3 x. 5 ) ) )
90	52, 89	breqtrrd 4681	. . 3 |- ( ( n e. NN /\ 3 || ( ( 4 ^ n ) + 5 ) ) -> 3 || ( ( 4 ^ ( n + 1 ) ) + 5 ) )
91	90	ex 450	. 2 |- ( n e. NN -> ( 3 || ( ( 4 ^ n ) + 5 ) -> 3 || ( ( 4 ^ ( n + 1 ) ) + 5 ) ) )
92	3, 6, 9, 12, 34, 91	nnind 11038	1 |- ( N e. NN -> 3 || ( ( 4 ^ N ) + 5 ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   NNcn 11020   3c3 11071   4c4 11072   5c5 11073   9c9 11077   NN0cn0 11292   ZZcz 11377  ;cdc 11493   ^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-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-seq 12802  df-exp 12861  df-dvds 14984

This theorem is referenced by: (None)