Theorem p1modz1 11690

Description: If a number greater than 1 divides another number, the second number increased by 1 is 1 modulo the first number. (Contributed by AV, 19-Mar-2022.)

Assertion

Ref	Expression
p1modz1	|- ( ( M || A /\ 1 < M ) -> ( ( A + 1 ) mod M ) = 1 )

Proof of Theorem p1modz1

Step	Hyp	Ref	Expression
1		dvdszrcl 11688	. . 3 |- ( M || A -> ( M e. ZZ /\ A e. ZZ ) )
2		0red 7879	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 0 e. RR )
3		1red 7893	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 1 e. RR )
4		zre 9171	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> M e. RR )
5	4	adantr 274	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> M e. RR )
6	2, 3, 5	3jca 1162	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 1 < M ) -> ( 0 e. RR /\ 1 e. RR /\ M e. RR ) )
7		0lt1 8002	. . . . . . . . . . . . . . 15 |- 0 < 1
8	7	a1i 9	. . . . . . . . . . . . . 14 |- ( M e. ZZ -> 0 < 1 )
9	8	anim1i 338	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 1 < M ) -> ( 0 < 1 /\ 1 < M ) )
10		lttr 7951	. . . . . . . . . . . . 13 |- ( ( 0 e. RR /\ 1 e. RR /\ M e. RR ) -> ( ( 0 < 1 /\ 1 < M ) -> 0 < M ) )
11	6, 9, 10	sylc 62	. . . . . . . . . . . 12 |- ( ( M e. ZZ /\ 1 < M ) -> 0 < M )
12	11	ex 114	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 1 < M -> 0 < M ) )
13		elnnz 9177	. . . . . . . . . . . 12 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
14	13	simplbi2 383	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 0 < M -> M e. NN ) )
15	12, 14	syld 45	. . . . . . . . . 10 |- ( M e. ZZ -> ( 1 < M -> M e. NN ) )
16	15	adantr 274	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( 1 < M -> M e. NN ) )
17	16	imp 123	. . . . . . . 8 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> M e. NN )
18		dvdsmod0 11689	. . . . . . . 8 |- ( ( M e. NN /\ M || A ) -> ( A mod M ) = 0 )
19	17, 18	sylan 281	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ M || A ) -> ( A mod M ) = 0 )
20	19	ex 114	. . . . . 6 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( M || A -> ( A mod M ) = 0 ) )
21		oveq1 5831	. . . . . . . . . . 11 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = ( 0 + 1 ) )
22		0p1e1 8947	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
23	21, 22	eqtrdi 2206	. . . . . . . . . 10 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = 1 )
24	23	oveq1d 5839	. . . . . . . . 9 |- ( ( A mod M ) = 0 -> ( ( ( A mod M ) + 1 ) mod M ) = ( 1 mod M ) )
25	24	adantl 275	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( 1 mod M ) )
26		zq 9535	. . . . . . . . . 10 |- ( A e. ZZ -> A e. QQ )
27	26	ad3antlr 485	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> A e. QQ )
28		1z 9193	. . . . . . . . . 10 |- 1 e. ZZ
29		zq 9535	. . . . . . . . . 10 |- ( 1 e. ZZ -> 1 e. QQ )
30	28, 29	mp1i 10	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> 1 e. QQ )
31		zq 9535	. . . . . . . . . 10 |- ( M e. ZZ -> M e. QQ )
32	31	ad3antrrr 484	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> M e. QQ )
33	11	ad4ant13 505	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> 0 < M )
34		modqaddmod 10262	. . . . . . . . 9 |- ( ( ( A e. QQ /\ 1 e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
35	27, 30, 32, 33, 34	syl22anc 1221	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
36	31	adantr 274	. . . . . . . . . 10 |- ( ( M e. ZZ /\ A e. ZZ ) -> M e. QQ )
37		q1mod 10255	. . . . . . . . . 10 |- ( ( M e. QQ /\ 1 < M ) -> ( 1 mod M ) = 1 )
38	36, 37	sylan 281	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( 1 mod M ) = 1 )
39	38	adantr 274	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( 1 mod M ) = 1 )
40	25, 35, 39	3eqtr3d 2198	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( ( A + 1 ) mod M ) = 1 )
41	40	ex 114	. . . . . 6 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( ( A mod M ) = 0 -> ( ( A + 1 ) mod M ) = 1 ) )
42	20, 41	syld 45	. . . . 5 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( M || A -> ( ( A + 1 ) mod M ) = 1 ) )
43	42	ex 114	. . . 4 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( 1 < M -> ( M || A -> ( ( A + 1 ) mod M ) = 1 ) ) )
44	43	com23 78	. . 3 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( M || A -> ( 1 < M -> ( ( A + 1 ) mod M ) = 1 ) ) )
45	1, 44	mpcom 36	. 2 |- ( M || A -> ( 1 < M -> ( ( A + 1 ) mod M ) = 1 ) )
46	45	imp 123	1 |- ( ( M || A /\ 1 < M ) -> ( ( A + 1 ) mod M ) = 1 )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 963   = wceq 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733   + caddc 7735   < clt 7912   NNcn 8833   ZZcz 9167   QQcq 9528   mod cmo 10221   || cdvds 11683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850  ax-arch 7851

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457  df-div 8546  df-inn 8834  df-n0 9091  df-z 9168  df-q 9529  df-rp 9561  df-fl 10169  df-mod 10222  df-dvds 11684

This theorem is referenced by: (None)