Theorem p1modz1 11940

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 11938	. . 3 |- ( M || A -> ( M e. ZZ /\ A e. ZZ ) )
2		0red 8022	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 0 e. RR )
3		1red 8036	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 1 e. RR )
4		zre 9324	. . . . . . . . . . . . . . 15 |- ( M e. ZZ -> M e. RR )
5	4	adantr 276	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> M e. RR )
6	2, 3, 5	3jca 1179	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 1 < M ) -> ( 0 e. RR /\ 1 e. RR /\ M e. RR ) )
7		0lt1 8148	. . . . . . . . . . . . . . 15 |- 0 < 1
8	7	a1i 9	. . . . . . . . . . . . . 14 |- ( M e. ZZ -> 0 < 1 )
9	8	anim1i 340	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 1 < M ) -> ( 0 < 1 /\ 1 < M ) )
10		lttr 8095	. . . . . . . . . . . . 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 115	. . . . . . . . . . 11 |- ( M e. ZZ -> ( 1 < M -> 0 < M ) )
13		elnnz 9330	. . . . . . . . . . . 12 |- ( M e. NN <-> ( M e. ZZ /\ 0 < M ) )
14	13	simplbi2 385	. . . . . . . . . . 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 276	. . . . . . . . 9 |- ( ( M e. ZZ /\ A e. ZZ ) -> ( 1 < M -> M e. NN ) )
17	16	imp 124	. . . . . . . 8 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> M e. NN )
18		dvdsmod0 11939	. . . . . . . 8 |- ( ( M e. NN /\ M || A ) -> ( A mod M ) = 0 )
19	17, 18	sylan 283	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ M || A ) -> ( A mod M ) = 0 )
20	19	ex 115	. . . . . 6 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( M || A -> ( A mod M ) = 0 ) )
21		oveq1 5926	. . . . . . . . . . 11 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = ( 0 + 1 ) )
22		0p1e1 9098	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
23	21, 22	eqtrdi 2242	. . . . . . . . . 10 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = 1 )
24	23	oveq1d 5934	. . . . . . . . 9 |- ( ( A mod M ) = 0 -> ( ( ( A mod M ) + 1 ) mod M ) = ( 1 mod M ) )
25	24	adantl 277	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( 1 mod M ) )
26		zq 9694	. . . . . . . . . 10 |- ( A e. ZZ -> A e. QQ )
27	26	ad3antlr 493	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> A e. QQ )
28		1z 9346	. . . . . . . . . 10 |- 1 e. ZZ
29		zq 9694	. . . . . . . . . 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 9694	. . . . . . . . . 10 |- ( M e. ZZ -> M e. QQ )
32	31	ad3antrrr 492	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> M e. QQ )
33	11	ad4ant13 513	. . . . . . . . 9 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> 0 < M )
34		modqaddmod 10437	. . . . . . . . 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 1250	. . . . . . . 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 276	. . . . . . . . . 10 |- ( ( M e. ZZ /\ A e. ZZ ) -> M e. QQ )
37		q1mod 10430	. . . . . . . . . 10 |- ( ( M e. QQ /\ 1 < M ) -> ( 1 mod M ) = 1 )
38	36, 37	sylan 283	. . . . . . . . 9 |- ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) -> ( 1 mod M ) = 1 )
39	38	adantr 276	. . . . . . . 8 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( 1 mod M ) = 1 )
40	25, 35, 39	3eqtr3d 2234	. . . . . . 7 |- ( ( ( ( M e. ZZ /\ A e. ZZ ) /\ 1 < M ) /\ ( A mod M ) = 0 ) -> ( ( A + 1 ) mod M ) = 1 )
41	40	ex 115	. . . . . 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 115	. . . 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 124	1 |- ( ( M || A /\ 1 < M ) -> ( ( A + 1 ) mod M ) = 1 )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   0cc0 7874   1c1 7875   + caddc 7877   < clt 8056   NNcn 8984   ZZcz 9320   QQcq 9687   mod cmo 10396   || cdvds 11933

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-n0 9244  df-z 9321  df-q 9688  df-rp 9723  df-fl 10342  df-mod 10397  df-dvds 11934

This theorem is referenced by:  lgslem4  15160