Theorem mulp1mod1 10727

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 9865	. . . . . . . . 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 9701	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. CC )
5	2, 4	mulcomd 8295	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) = ( A x. N ) )
6	5	oveq1d 6065	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = ( ( A x. N ) mod N ) )
7		eluzelz 9863	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
8		zq 9958	. . . . . . . . 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 8275	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 e. RR )
12		2re 9307	. . . . . . . . 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 9700	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. RR )
16		2pos 9328	. . . . . . . . 9 |- 0 < 2
17	16	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < 2 )
18		eluzle 9866	. . . . . . . . 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 8697	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < N )
21		mulqmod0 10692	. . . . . . 7 |- ( ( A e. ZZ /\ N e. QQ /\ 0 < N ) -> ( ( A x. N ) mod N ) = 0 )
22	3, 10, 20, 21	syl3anc 1274	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A x. N ) mod N ) = 0 )
23	6, 22	eqtrd 2265	. . . . 5 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = 0 )
24	23	oveq1d 6065	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = ( 0 + 1 ) )
25		0p1e1 9351	. . . 4 |- ( 0 + 1 ) = 1
26	24, 25	eqtrdi 2281	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = 1 )
27	26	oveq1d 6065	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( 1 mod N ) )
28		zq 9958	. . . . 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 9969	. . . 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 9603	. . . 4 |- 1 e. ZZ
33		zq 9958	. . . 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 10725	. . 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 1275	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
37		eluz2gt1 9934	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
38	37	adantl 277	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 < N )
39		q1mod 10718	. . 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 2273	1 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   <_ cle 8309   2c2 9288   ZZcz 9577   ZZ>=cuz 9853   QQcq 9951   mod cmo 10684

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685

This theorem is referenced by: (None)