Theorem mulp1mod1 10626

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 9766	. . . . . . . . 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 9602	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. CC )
5	2, 4	mulcomd 8200	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) = ( A x. N ) )
6	5	oveq1d 6032	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = ( ( A x. N ) mod N ) )
7		eluzelz 9764	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
8		zq 9859	. . . . . . . . 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 8179	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 e. RR )
12		2re 9212	. . . . . . . . 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 9601	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. RR )
16		2pos 9233	. . . . . . . . 9 |- 0 < 2
17	16	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < 2 )
18		eluzle 9767	. . . . . . . . 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 8602	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < N )
21		mulqmod0 10591	. . . . . . 7 |- ( ( A e. ZZ /\ N e. QQ /\ 0 < N ) -> ( ( A x. N ) mod N ) = 0 )
22	3, 10, 20, 21	syl3anc 1273	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A x. N ) mod N ) = 0 )
23	6, 22	eqtrd 2264	. . . . 5 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = 0 )
24	23	oveq1d 6032	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = ( 0 + 1 ) )
25		0p1e1 9256	. . . 4 |- ( 0 + 1 ) = 1
26	24, 25	eqtrdi 2280	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = 1 )
27	26	oveq1d 6032	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( 1 mod N ) )
28		zq 9859	. . . . 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 9870	. . . 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 9504	. . . 4 |- 1 e. ZZ
33		zq 9859	. . . 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 10624	. . 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 1274	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
37		eluz2gt1 9835	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
38	37	adantl 277	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 < N )
39		q1mod 10617	. . 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 2272	1 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   <_ cle 8214   2c2 9193   ZZcz 9478   ZZ>=cuz 9754   QQcq 9852   mod cmo 10583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fl 10529  df-mod 10584

This theorem is referenced by: (None)