Theorem mod42tp1mod8 41519

Description: If a number is 3 modulo 4, twice the number plus 1 is 7 modulo 8. (Contributed by AV, 19-Aug-2021.)

Assertion

Ref	Expression
mod42tp1mod8	|- ( ( N e. ZZ /\ ( N mod 4 ) = 3 ) -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 )

Proof of Theorem mod42tp1mod8

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		4nn 11187	. . . . 5 |- 4 e. NN
2	1	a1i 11	. . . 4 |- ( N e. ZZ -> 4 e. NN )
3		3nn0 11310	. . . . . 6 |- 3 e. NN0
4	3	a1i 11	. . . . 5 |- ( N e. ZZ -> 3 e. NN0 )
5		3lt4 11197	. . . . 5 |- 3 < 4
6	4, 5	jctir 561	. . . 4 |- ( N e. ZZ -> ( 3 e. NN0 /\ 3 < 4 ) )
7		modremain 15132	. . . 4 |- ( ( N e. ZZ /\ 4 e. NN /\ ( 3 e. NN0 /\ 3 < 4 ) ) -> ( ( N mod 4 ) = 3 <-> E. z e. ZZ ( ( z x. 4 ) + 3 ) = N ) )
8	2, 6, 7	mpd3an23 1426	. . 3 |- ( N e. ZZ -> ( ( N mod 4 ) = 3 <-> E. z e. ZZ ( ( z x. 4 ) + 3 ) = N ) )
9		2cnd 11093	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ z e. ZZ ) -> 2 e. CC )
10		simpr 477	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ z e. ZZ ) -> z e. ZZ )
11		4z 11411	. . . . . . . . . . . . . 14 |- 4 e. ZZ
12	11	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ z e. ZZ ) -> 4 e. ZZ )
13	10, 12	zmulcld 11488	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( z x. 4 ) e. ZZ )
14	13	zcnd 11483	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( z x. 4 ) e. CC )
15		3cn 11095	. . . . . . . . . . . 12 |- 3 e. CC
16	15	a1i 11	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ z e. ZZ ) -> 3 e. CC )
17	9, 14, 16	adddid 10064	. . . . . . . . . 10 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( ( 2 x. ( z x. 4 ) ) + ( 2 x. 3 ) ) )
18	10	zcnd 11483	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ z e. ZZ ) -> z e. CC )
19		4cn 11098	. . . . . . . . . . . . . 14 |- 4 e. CC
20	19	a1i 11	. . . . . . . . . . . . 13 |- ( ( N e. ZZ /\ z e. ZZ ) -> 4 e. CC )
21	9, 18, 20	mul12d 10245	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( z x. 4 ) ) = ( z x. ( 2 x. 4 ) ) )
22		2cn 11091	. . . . . . . . . . . . . 14 |- 2 e. CC
23		4t2e8 11181	. . . . . . . . . . . . . 14 |- ( 4 x. 2 ) = 8
24	19, 22, 23	mulcomli 10047	. . . . . . . . . . . . 13 |- ( 2 x. 4 ) = 8
25	24	oveq2i 6661	. . . . . . . . . . . 12 |- ( z x. ( 2 x. 4 ) ) = ( z x. 8 )
26	21, 25	syl6eq 2672	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( z x. 4 ) ) = ( z x. 8 ) )
27		3t2e6 11179	. . . . . . . . . . . . 13 |- ( 3 x. 2 ) = 6
28	15, 22, 27	mulcomli 10047	. . . . . . . . . . . 12 |- ( 2 x. 3 ) = 6
29	28	a1i 11	. . . . . . . . . . 11 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. 3 ) = 6 )
30	26, 29	oveq12d 6668	. . . . . . . . . 10 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( z x. 4 ) ) + ( 2 x. 3 ) ) = ( ( z x. 8 ) + 6 ) )
31	17, 30	eqtrd 2656	. . . . . . . . 9 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( ( z x. 8 ) + 6 ) )
32	31	oveq1d 6665	. . . . . . . 8 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( ( z x. 8 ) + 6 ) + 1 ) )
33		id 22	. . . . . . . . . . . . 13 |- ( z e. ZZ -> z e. ZZ )
34		8nn 11191	. . . . . . . . . . . . . . 15 |- 8 e. NN
35	34	nnzi 11401	. . . . . . . . . . . . . 14 |- 8 e. ZZ
36	35	a1i 11	. . . . . . . . . . . . 13 |- ( z e. ZZ -> 8 e. ZZ )
37	33, 36	zmulcld 11488	. . . . . . . . . . . 12 |- ( z e. ZZ -> ( z x. 8 ) e. ZZ )
38	37	zcnd 11483	. . . . . . . . . . 11 |- ( z e. ZZ -> ( z x. 8 ) e. CC )
39		6cn 11102	. . . . . . . . . . . 12 |- 6 e. CC
40	39	a1i 11	. . . . . . . . . . 11 |- ( z e. ZZ -> 6 e. CC )
41		1cnd 10056	. . . . . . . . . . 11 |- ( z e. ZZ -> 1 e. CC )
42	38, 40, 41	addassd 10062	. . . . . . . . . 10 |- ( z e. ZZ -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + ( 6 + 1 ) ) )
43		6p1e7 11156	. . . . . . . . . . . 12 |- ( 6 + 1 ) = 7
44	43	a1i 11	. . . . . . . . . . 11 |- ( z e. ZZ -> ( 6 + 1 ) = 7 )
45	44	oveq2d 6666	. . . . . . . . . 10 |- ( z e. ZZ -> ( ( z x. 8 ) + ( 6 + 1 ) ) = ( ( z x. 8 ) + 7 ) )
46	42, 45	eqtrd 2656	. . . . . . . . 9 |- ( z e. ZZ -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + 7 ) )
47	46	adantl 482	. . . . . . . 8 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 8 ) + 6 ) + 1 ) = ( ( z x. 8 ) + 7 ) )
48	32, 47	eqtrd 2656	. . . . . . 7 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( z x. 8 ) + 7 ) )
49	48	oveq1d 6665	. . . . . 6 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = ( ( ( z x. 8 ) + 7 ) mod 8 ) )
50		nnrp 11842	. . . . . . . . 9 |- ( 8 e. NN -> 8 e. RR+ )
51	34, 50	mp1i 13	. . . . . . . 8 |- ( z e. ZZ -> 8 e. RR+ )
52		0xr 10086	. . . . . . . . . 10 |- 0 e. RR*
53	52	a1i 11	. . . . . . . . 9 |- ( z e. ZZ -> 0 e. RR* )
54		8re 11105	. . . . . . . . . . 11 |- 8 e. RR
55	54	rexri 10097	. . . . . . . . . 10 |- 8 e. RR*
56	55	a1i 11	. . . . . . . . 9 |- ( z e. ZZ -> 8 e. RR* )
57		7re 11103	. . . . . . . . . . 11 |- 7 e. RR
58	57	rexri 10097	. . . . . . . . . 10 |- 7 e. RR*
59	58	a1i 11	. . . . . . . . 9 |- ( z e. ZZ -> 7 e. RR* )
60		0re 10040	. . . . . . . . . . 11 |- 0 e. RR
61		7pos 11120	. . . . . . . . . . 11 |- 0 < 7
62	60, 57, 61	ltleii 10160	. . . . . . . . . 10 |- 0 <_ 7
63	62	a1i 11	. . . . . . . . 9 |- ( z e. ZZ -> 0 <_ 7 )
64		7lt8 11215	. . . . . . . . . 10 |- 7 < 8
65	64	a1i 11	. . . . . . . . 9 |- ( z e. ZZ -> 7 < 8 )
66	53, 56, 59, 63, 65	elicod 12224	. . . . . . . 8 |- ( z e. ZZ -> 7 e. ( 0 [,) 8 ) )
67		muladdmodid 12710	. . . . . . . 8 |- ( ( z e. ZZ /\ 8 e. RR+ /\ 7 e. ( 0 [,) 8 ) ) -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 )
68	51, 66, 67	mpd3an23 1426	. . . . . . 7 |- ( z e. ZZ -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 )
69	68	adantl 482	. . . . . 6 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 8 ) + 7 ) mod 8 ) = 7 )
70	49, 69	eqtrd 2656	. . . . 5 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = 7 )
71		oveq2 6658	. . . . . . . 8 |- ( ( ( z x. 4 ) + 3 ) = N -> ( 2 x. ( ( z x. 4 ) + 3 ) ) = ( 2 x. N ) )
72	71	oveq1d 6665	. . . . . . 7 |- ( ( ( z x. 4 ) + 3 ) = N -> ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) = ( ( 2 x. N ) + 1 ) )
73	72	oveq1d 6665	. . . . . 6 |- ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = ( ( ( 2 x. N ) + 1 ) mod 8 ) )
74	73	eqeq1d 2624	. . . . 5 |- ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( ( 2 x. ( ( z x. 4 ) + 3 ) ) + 1 ) mod 8 ) = 7 <-> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) )
75	70, 74	syl5ibcom 235	. . . 4 |- ( ( N e. ZZ /\ z e. ZZ ) -> ( ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) )
76	75	rexlimdva 3031	. . 3 |- ( N e. ZZ -> ( E. z e. ZZ ( ( z x. 4 ) + 3 ) = N -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) )
77	8, 76	sylbid 230	. 2 |- ( N e. ZZ -> ( ( N mod 4 ) = 3 -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 ) )
78	77	imp 445	1 |- ( ( N e. ZZ /\ ( N mod 4 ) = 3 ) -> ( ( ( 2 x. N ) + 1 ) mod 8 ) = 7 )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   3c3 11071   4c4 11072   6c6 11074   7c7 11075   8c8 11076   NN0cn0 11292   ZZcz 11377   RR+crp 11832   [,)cico 12177   mod cmo 12668

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  ax-pre-sup 10014

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-rmo 2920  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-1st 7168  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-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984

This theorem is referenced by:  sgprmdvdsmersenne  41521