Theorem addmodid 10389

Description: The sum of a positive integer and a nonnegative integer less than the positive integer is equal to the nonnegative integer modulo the positive integer. (Contributed by Alexander van der Vekens, 30-Oct-2018.) (Proof shortened by AV, 5-Jul-2020.)

Assertion

Ref	Expression
addmodid	|- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = A )

Proof of Theorem addmodid

Step	Hyp	Ref	Expression
1		simp2 999	. . . . . . 7 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> M e. NN )
2	1	nncnd 8950	. . . . . 6 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> M e. CC )
3	2	mulid2d 7993	. . . . 5 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( 1 x. M ) = M )
4	3	eqcomd 2194	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> M = ( 1 x. M ) )
5	4	oveq1d 5905	. . 3 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( M + A ) = ( ( 1 x. M ) + A ) )
6	5	oveq1d 5905	. 2 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = ( ( ( 1 x. M ) + A ) mod M ) )
7		1zzd 9297	. . 3 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> 1 e. ZZ )
8		nnq 9650	. . . 4 |- ( M e. NN -> M e. QQ )
9	8	3ad2ant2 1020	. . 3 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> M e. QQ )
10		simp1 998	. . . . 5 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A e. NN0 )
11	10	nn0zd 9390	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A e. ZZ )
12		zq 9643	. . . 4 |- ( A e. ZZ -> A e. QQ )
13	11, 12	syl 14	. . 3 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A e. QQ )
14		nn0re 9202	. . . . 5 |- ( A e. NN0 -> A e. RR )
15	14	3ad2ant1 1019	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A e. RR )
16	10	nn0ge0d 9249	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> 0 <_ A )
17		simp3 1000	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A < M )
18		0re 7974	. . . . 5 |- 0 e. RR
19		nnre 8943	. . . . . . 7 |- ( M e. NN -> M e. RR )
20	19	rexrd 8024	. . . . . 6 |- ( M e. NN -> M e. RR* )
21	20	3ad2ant2 1020	. . . . 5 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> M e. RR* )
22		elico2 9954	. . . . 5 |- ( ( 0 e. RR /\ M e. RR* ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
23	18, 21, 22	sylancr 414	. . . 4 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
24	15, 16, 17, 23	mpbir3and 1181	. . 3 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> A e. ( 0 [,) M ) )
25		mulqaddmodid 10381	. . 3 |- ( ( ( 1 e. ZZ /\ M e. QQ ) /\ ( A e. QQ /\ A e. ( 0 [,) M ) ) ) -> ( ( ( 1 x. M ) + A ) mod M ) = A )
26	7, 9, 13, 24, 25	syl22anc 1249	. 2 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( ( 1 x. M ) + A ) mod M ) = A )
27	6, 26	eqtrd 2221	1 |- ( ( A e. NN0 /\ M e. NN /\ A < M ) -> ( ( M + A ) mod M ) = A )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2159   class class class wbr 4017  (class class class)co 5890   RRcr 7827   0cc0 7828   1c1 7829   + caddc 7831   x. cmul 7833   RR*cxr 8008   < clt 8009   <_ cle 8010   NNcn 8936   NN0cn0 9193   ZZcz 9270   QQcq 9636   [,)cico 9907   mod cmo 10339

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 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447  ax-setind 4550  ax-cnex 7919  ax-resscn 7920  ax-1cn 7921  ax-1re 7922  ax-icn 7923  ax-addcl 7924  ax-addrcl 7925  ax-mulcl 7926  ax-mulrcl 7927  ax-addcom 7928  ax-mulcom 7929  ax-addass 7930  ax-mulass 7931  ax-distr 7932  ax-i2m1 7933  ax-0lt1 7934  ax-1rid 7935  ax-0id 7936  ax-rnegex 7937  ax-precex 7938  ax-cnre 7939  ax-pre-ltirr 7940  ax-pre-ltwlin 7941  ax-pre-lttrn 7942  ax-pre-apti 7943  ax-pre-ltadd 7944  ax-pre-mulgt0 7945  ax-pre-mulext 7946  ax-arch 7947

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 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ne 2360  df-nel 2455  df-ral 2472  df-rex 2473  df-reu 2474  df-rmo 2475  df-rab 2476  df-v 2753  df-sbc 2977  df-csb 3072  df-dif 3145  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-int 3859  df-iun 3902  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-po 4310  df-iso 4311  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-rn 4651  df-res 4652  df-ima 4653  df-iota 5192  df-fun 5232  df-fn 5233  df-f 5234  df-fv 5238  df-riota 5846  df-ov 5893  df-oprab 5894  df-mpo 5895  df-1st 6158  df-2nd 6159  df-pnf 8011  df-mnf 8012  df-xr 8013  df-ltxr 8014  df-le 8015  df-sub 8147  df-neg 8148  df-reap 8549  df-ap 8556  df-div 8647  df-inn 8937  df-n0 9194  df-z 9271  df-q 9637  df-rp 9671  df-ico 9911  df-fl 10287  df-mod 10340

This theorem is referenced by:  addmodidr  10390