Theorem p1modz1 12305

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 12303	. . 3 |- ( M || A -> ( M e. ZZ /\ A e. ZZ ) )
2		0red 8147	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 0 e. RR )
3		1red 8161	. . . . . . . . . . . . . 14 |- ( ( M e. ZZ /\ 1 < M ) -> 1 e. RR )
4		zre 9450	. . . . . . . . . . . . . . 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 1201	. . . . . . . . . . . . 13 |- ( ( M e. ZZ /\ 1 < M ) -> ( 0 e. RR /\ 1 e. RR /\ M e. RR ) )
7		0lt1 8273	. . . . . . . . . . . . . . 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 8220	. . . . . . . . . . . . 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 9456	. . . . . . . . . . . 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 12304	. . . . . . . 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 6008	. . . . . . . . . . 11 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = ( 0 + 1 ) )
22		0p1e1 9224	. . . . . . . . . . 11 |- ( 0 + 1 ) = 1
23	21, 22	eqtrdi 2278	. . . . . . . . . 10 |- ( ( A mod M ) = 0 -> ( ( A mod M ) + 1 ) = 1 )
24	23	oveq1d 6016	. . . . . . . . 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 9821	. . . . . . . . . 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 9472	. . . . . . . . . 10 |- 1 e. ZZ
29		zq 9821	. . . . . . . . . 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 9821	. . . . . . . . . 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 10585	. . . . . . . . 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 1272	. . . . . . . 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 10578	. . . . . . . . . 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 2270	. . . . . . 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   < clt 8181   NNcn 9110   ZZcz 9446   QQcq 9814   mod cmo 10544   || cdvds 12298

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-n0 9370  df-z 9447  df-q 9815  df-rp 9850  df-fl 10490  df-mod 10545  df-dvds 12299

This theorem is referenced by:  lgslem4  15682