Theorem mulp1mod1 10823

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 9949	. . . . . . . . 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 9780	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> A e. CC )
5	2, 4	mulcomd 8353	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) = ( A x. N ) )
6	5	oveq1d 6103	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = ( ( A x. N ) mod N ) )
7		eluzelz 9947	. . . . . . . . 9 |- ( N e. ( ZZ>= ` 2 ) -> N e. ZZ )
8		zq 10042	. . . . . . . . 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 8333	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 e. RR )
12		2re 9383	. . . . . . . . 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 9779	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> N e. RR )
16		2pos 9404	. . . . . . . . 9 |- 0 < 2
17	16	a1i 9	. . . . . . . 8 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < 2 )
18		eluzle 9950	. . . . . . . . 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 8758	. . . . . . 7 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 0 < N )
21		mulqmod0 10788	. . . . . . 7 |- ( ( A e. ZZ /\ N e. QQ /\ 0 < N ) -> ( ( A x. N ) mod N ) = 0 )
22	3, 10, 20, 21	syl3anc 1278	. . . . . 6 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( A x. N ) mod N ) = 0 )
23	6, 22	eqtrd 2271	. . . . 5 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( N x. A ) mod N ) = 0 )
24	23	oveq1d 6103	. . . 4 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = ( 0 + 1 ) )
25		0p1e1 9427	. . . 4 |- ( 0 + 1 ) = 1
26	24, 25	eqtrdi 2287	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) mod N ) + 1 ) = 1 )
27	26	oveq1d 6103	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( 1 mod N ) )
28		zq 10042	. . . . 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 10053	. . . 4 |- ( ( N e. QQ /\ A e. QQ ) -> ( N x. A ) e. QQ )
31	10, 29, 30	syl2anc 415	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( N x. A ) e. QQ )
32		1z 9681	. . . 4 |- 1 e. ZZ
33		zq 10042	. . . 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 10821	. . 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 1279	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( ( N x. A ) mod N ) + 1 ) mod N ) = ( ( ( N x. A ) + 1 ) mod N ) )
37		eluz2gt1 10018	. . . 4 |- ( N e. ( ZZ>= ` 2 ) -> 1 < N )
38	37	adantl 277	. . 3 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> 1 < N )
39		q1mod 10814	. . 3 |- ( ( N e. QQ /\ 1 < N ) -> ( 1 mod N ) = 1 )
40	10, 38, 39	syl2anc 415	. 2 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( 1 mod N ) = 1 )
41	27, 36, 40	3eqtr3d 2279	1 |- ( ( A e. ZZ /\ N e. ( ZZ>= ` 2 ) ) -> ( ( ( N x. A ) + 1 ) mod N ) = 1 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   2c2 9364   ZZcz 9655   ZZ>=cuz 9937   QQcq 10035   mod cmo 10780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fl 10722  df-mod 10781

This theorem is used by: (None)