Theorem mulp1mod1 10439

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 9606	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. CC )
2	1	adantl 277	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. CC )
3		simpl 109	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. ZZ )
4	3	zcnd 9443	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. CC )
5	2, 4	mulcomd 8043	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) = ( A x. N ) )
6	5	oveq1d 5934	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = ( ( A x. N ) mod N ) )
7		eluzelz 9604	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
8		zq 9694	. . . . . . . . 9 |- ( N e. ZZ -> N e. QQ )
9	7, 8	syl 14	. . . . . . . 8 |- ( N e. ( ZZ>= ` 2 ) -> N e. QQ )
10	9	adantl 277	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. QQ )
11		0red 8022	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 e. RR )
12		2re 9054	. . . . . . . . 9 |- 2 e. RR
13	12	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 2 e. RR )
14	7	adantl 277	. . . . . . . . 9 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. ZZ )
15	14	zred 9442	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. RR )
16		2pos 9075	. . . . . . . . 9 |- 0 < 2
17	16	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < 2 )
18		eluzle 9607	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> 2 <_ N )
19	18	adantl 277	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 2 <_ N )
20	11, 13, 15, 17, 19	ltletrd 8444	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < N )
21		mulqmod0 10404	. . . . . . 7 |- ( ( A e. ZZ /\ N e. QQ /\ 0 < N ) -> ( ( A x. N ) mod N ) = 0 )
22	3, 10, 20, 21	syl3anc 1249	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A x. N ) mod N ) = 0 )
23	6, 22	eqtrd 2226	. . . . 5 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = 0 )
24	23	oveq1d 5934	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = ( 0 + 1 ) )
25		0p1e1 9098	. . . 4 |- ( 0 + 1 ) = 1
26	24, 25	eqtrdi 2242	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = 1 )
27	26	oveq1d 5934	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( 1 mod N ) )
28		zq 9694	. . . . 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 9705	. . . 4 |- ( ( N e. QQ /\ A e. QQ ) -> ( N x. A ) e. QQ )
31	10, 29, 30	syl2anc 411	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) e. QQ )
32		1z 9346	. . . 4 |- 1 e. ZZ
33		zq 9694	. . . 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 10437	. . 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 1250	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
37		eluz2gt1 9670	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
38	37	adantl 277	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 < N )
39		q1mod 10430	. . 3 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
40	10, 38, 39	syl2anc 411	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( 1 mod N ) = 1 )
41	27, 36, 40	3eqtr3d 2234	1 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   x. cmul 7879   < clt 8056   <_ cle 8057   2c2 9035   ZZcz 9320   ZZ>=cuz 9595   QQcq 9687   mod cmo 10396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fl 10342  df-mod 10397

This theorem is referenced by: (None)