Theorem modqltm1p1mod 10389

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 9292	. . . 4 |- 1 e. ZZ
3		zq 9639	. . . 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 10376	. . 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 1249	. 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 10341	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( A mod M ) e. QQ )
10		qaddcl 9648	. . . 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 7971	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 e. RR )
13		qre 9638	. . . . 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 7985	. . . . 5 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 1 e. RR )
16	14, 15	readdcld 8000	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> ( ( A mod M ) + 1 ) e. RR )
17		modqge0 10345	. . . . 5 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
18	1, 5, 6, 17	syl3anc 1248	. . . 4 |- ( ( ( A e. QQ /\ ( A mod M ) < ( M - 1 ) ) /\ ( M e. QQ /\ 0 < M ) ) -> 0 <_ ( A mod M ) )
19	14	lep1d 8901	. . . 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 8094	. . 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 9638	. . . . . 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 8515	. . . 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 10362	. . 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 1249	. 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 2222	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 1363   e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825   + caddc 7827   < clt 8005   <_ cle 8006   - cmin 8141   ZZcz 9266   QQcq 9632   mod cmo 10335

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-n0 9190  df-z 9267  df-q 9633  df-rp 9667  df-fl 10283  df-mod 10336

This theorem is referenced by: (None)