Theorem modqm1p1mod0 10761

Description: If a number modulo a modulus equals the modulus decreased by 1, the first number increased by 1 modulo the modulus equals 0. (Contributed by Jim Kingdon, 24-Oct-2021.)

Assertion

Ref	Expression
modqm1p1mod0	|- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )

Proof of Theorem modqm1p1mod0

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> A e. QQ )
2		1z 9620	. . . . 5 |- 1 e. ZZ
3		zq 9976	. . . . 5 |- ( 1 e. ZZ -> 1 e. QQ )
4	2, 3	mp1i 10	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> 1 e. QQ )
5		simp2 1025	. . . . 5 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> M e. QQ )
6	5	adantr 276	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> M e. QQ )
7		simpl3 1029	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> 0 < M )
8		modqaddmod 10749	. . . 4 |- ( ( ( A e. QQ /\ 1 e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
9	1, 4, 6, 7, 8	syl22anc 1275	. . 3 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
10		oveq1 6065	. . . . . 6 |- ( ( A mod M ) = ( M - 1 ) -> ( ( A mod M ) + 1 ) = ( ( M - 1 ) + 1 ) )
11	10	oveq1d 6073	. . . . 5 |- ( ( A mod M ) = ( M - 1 ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( ( M - 1 ) + 1 ) mod M ) )
12	11	adantl 277	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( ( M - 1 ) + 1 ) mod M ) )
13		qcn 9984	. . . . . . . 8 |- ( M e. QQ -> M e. CC )
14	5, 13	syl 14	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> M e. CC )
15	14	adantr 276	. . . . . 6 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> M e. CC )
16		npcan1 8668	. . . . . 6 |- ( M e. CC -> ( ( M - 1 ) + 1 ) = M )
17	15, 16	syl 14	. . . . 5 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( M - 1 ) + 1 ) = M )
18	17	oveq1d 6073	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( ( M - 1 ) + 1 ) mod M ) = ( M mod M ) )
19		modqid0 10736	. . . . 5 |- ( ( M e. QQ /\ 0 < M ) -> ( M mod M ) = 0 )
20	6, 7, 19	syl2anc 411	. . . 4 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( M mod M ) = 0 )
21	12, 18, 20	3eqtrd 2271	. . 3 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = 0 )
22	9, 21	eqtr3d 2269	. 2 |- ( ( ( A e. QQ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = ( M - 1 ) ) -> ( ( A + 1 ) mod M ) = 0 )
23	22	ex 115	1 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = ( M - 1 ) -> ( ( A + 1 ) mod M ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   < clt 8324   - cmin 8460   ZZcz 9594   QQcq 9969   mod cmo 10708

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-q 9970  df-rp 10005  df-fl 10654  df-mod 10709

This theorem is referenced by: (None)