Theorem mulp1mod1 9499

Description: The product of an integer and an integer greater than 1 increased by 1 is 1 modulo the integer greater than 1. (Contributed by AV, 15-Jul-2021.)

Assertion

Ref	Expression
mulp1mod1	|- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

Proof of Theorem mulp1mod1

Step	Hyp	Ref	Expression
1		eluzelcn 8763	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. CC )
2	1	adantl 271	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. CC )
3		simpl 107	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. ZZ )
4	3	zcnd 8603	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. CC )
5	2, 4	mulcomd 7254	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) = ( A x. N ) )
6	5	oveq1d 5578	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = ( ( A x. N ) mod N ) )
7		eluzelz 8761	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
8		zq 8844	. . . . . . . . 9 |- ( N e. ZZ -> N e. QQ )
9	7, 8	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. QQ )
10	9	adantl 271	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. QQ )
11		0red 7234	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 e. RR )
12		2re 8228	. . . . . . . . 9 |- 2 e. RR
13	12	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 2 e. RR )
14	7	adantl 271	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. ZZ )
15	14	zred 8602	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. RR )
16		2pos 8249	. . . . . . . . 9 |- 0 < 2
17	16	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < 2 )
18		eluzle 8764	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> 2 <_ N )
19	18	adantl 271	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 2 <_ N )
20	11, 13, 15, 17, 19	ltletrd 7646	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < N )
21		mulqmod0 9464	. . . . . . 7 |- ( ( A e. ZZ /\ N e. QQ /\ 0 < N ) -> ( ( A x. N ) mod N ) = 0 )
22	3, 10, 20, 21	syl3anc 1170	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A x. N ) mod N ) = 0 )
23	6, 22	eqtrd 2115	. . . . 5 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = 0 )
24	23	oveq1d 5578	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = ( 0 + 1 ) )
25		0p1e1 8272	. . . 4 |- ( 0 + 1 ) = 1
26	24, 25	syl6eq 2131	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = 1 )
27	26	oveq1d 5578	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( 1 mod N ) )
28		zq 8844	. . . . 5 |- ( A e. ZZ -> A e. QQ )
29	3, 28	syl 14	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. QQ )
30		qmulcl 8855	. . . 4 |- ( ( N e. QQ /\ A e. QQ ) -> ( N x. A ) e. QQ )
31	10, 29, 30	syl2anc 403	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) e. QQ )
32		1z 8510	. . . 4 |- 1 e. ZZ
33		zq 8844	. . . 4 |- ( 1 e. ZZ -> 1 e. QQ )
34	32, 33	mp1i 10	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 e. QQ )
35		modqaddmod 9497	. . 3 |- ( ( ( ( N x. A ) e. QQ /\ 1 e. QQ ) /\ ( N e. QQ /\ 0 < N ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
36	31, 34, 10, 20, 35	syl22anc 1171	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
37		eluz2gt1 8822	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
38	37	adantl 271	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 < N )
39		q1mod 9490	. . 3 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
40	10, 38, 39	syl2anc 403	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( 1 mod N ) = 1 )
41	27, 36, 40	3eqtr3d 2123	1 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   class class class wbr 3805   ` cfv 4952  (class class class)co 5563   CCcc 7093   RRcr 7094   0cc0 7095   1c1 7096   + caddc 7098   x. cmul 7100   < clt 7267   <_ cle 7268   2c2 8208   ZZcz 8484   ZZ>=cuz 8752   QQcq 8837   mod cmo 9456

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fl 9404  df-mod 9457

This theorem is referenced by: (None)