Theorem modqltm1p1mod 10468

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

Assertion

Ref	Expression
modqltm1p1mod	|- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) )

Proof of Theorem modqltm1p1mod

Step	Hyp	Ref	Expression
1		simpll 527	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> A e. QQ )
2		1z 9352	. . . 4 |- 1 e. ZZ
3		zq 9700	. . . 4 |- ( 1 e. ZZ -> 1 e. QQ )
4	2, 3	mp1i 10	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 1 e. QQ )
5		simprl 529	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. QQ )
6		simprr 531	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 < M )
7		modqaddmod 10455	. . 3 |- ( ( ( A e. QQ /\ 1 e. QQ ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
8	1, 4, 5, 6, 7	syl22anc 1250	. 2 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A + 1 ) mod M ) )
9	1, 5, 6	modqcld 10420	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A mod M ) e. QQ )
10		qaddcl 9709	. . . 4 |- ( ( ( A mod M ) e. QQ /\ 1 e. QQ ) -> ( ( A mod M ) + 1 ) e. QQ )
11	9, 4, 10	syl2anc 411	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A mod M ) + 1 ) e. QQ )
12		0red 8027	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 e. RR )
13		qre 9699	. . . . 5 |- ( ( A mod M ) e. QQ -> ( A mod M ) e. RR )
14	9, 13	syl 14	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A mod M ) e. RR )
15		1red 8041	. . . . 5 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 1 e. RR )
16	14, 15	readdcld 8056	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A mod M ) + 1 ) e. RR )
17		modqge0 10424	. . . . 5 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
18	1, 5, 6, 17	syl3anc 1249	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 <_ ( A mod M ) )
19	14	lep1d 8958	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A mod M ) <_ ( ( A mod M ) + 1 ) )
20	12, 14, 16, 18, 19	letrd 8150	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 <_ ( ( A mod M ) + 1 ) )
21		simplr 528	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A mod M ) < ( M - 1 ) )
22		qre 9699	. . . . . 6 |- ( M e. QQ -> M e. RR )
23	5, 22	syl 14	. . . . 5 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> M e. RR )
24	14, 15, 23	ltaddsubd 8572	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) < M <-> ( A mod M ) < ( M - 1 ) ) )
25	21, 24	mpbird 167	. . 3 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A mod M ) + 1 ) < M )
26		modqid 10441	. . 3 |- ( ( ( ( ( A mod M ) + 1 ) e. QQ /\ M e. QQ ) /\ ( 0 <_ ( ( A mod M ) + 1 ) /\ ( ( A mod M ) + 1 ) < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A mod M ) + 1 ) )
27	11, 5, 20, 25, 26	syl22anc 1250	. 2 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( ( A mod M ) + 1 ) mod M ) = ( ( A mod M ) + 1 ) )
28	8, 27	eqtr3d 2231	1 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A + 1 ) mod M ) = ( ( A mod M ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   < clt 8061   <_ cle 8062   - cmin 8197   ZZcz 9326   QQcq 9693   mod cmo 10414

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-po 4331  df-iso 4332  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-q 9694  df-rp 9729  df-fl 10360  df-mod 10415

This theorem is referenced by: (None)