Theorem muladdmodid 12710

Description: The sum of a positive real number less than an upper bound and the product of an integer and the upper bound is the positive real number modulo the upper bound. (Contributed by AV, 5-Jul-2020.)

Assertion

Ref	Expression
muladdmodid	|- ( ( N e. ZZ /\ M e. RR+ /\ A e. ( 0 [,) M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )

Proof of Theorem muladdmodid

Step	Hyp	Ref	Expression
1		0red 10041	. . . . 5 |- ( M e. RR+ -> 0 e. RR )
2		rpxr 11840	. . . . 5 |- ( M e. RR+ -> M e. RR* )
3		elico2 12237	. . . . 5 |- ( ( 0 e. RR /\ M e. RR* ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
4	1, 2, 3	syl2anc 693	. . . 4 |- ( M e. RR+ -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
5	4	adantl 482	. . 3 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
6		zcn 11382	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
7		rpcn 11841	. . . . . . . . 9 |- ( M e. RR+ -> M e. CC )
8		mulcl 10020	. . . . . . . . 9 |- ( ( N e. CC /\ M e. CC ) -> ( N x. M ) e. CC )
9	6, 7, 8	syl2an 494	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( N x. M ) e. CC )
10	9	adantr 481	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( N x. M ) e. CC )
11		recn 10026	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
12	11	3ad2ant1 1082	. . . . . . . 8 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> A e. CC )
13	12	adantl 482	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> A e. CC )
14	10, 13	addcomd 10238	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( ( N x. M ) + A ) = ( A + ( N x. M ) ) )
15	14	oveq1d 6665	. . . . 5 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( ( ( N x. M ) + A ) mod M ) = ( ( A + ( N x. M ) ) mod M ) )
16		simp1 1061	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> A e. RR )
17	16	adantl 482	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> A e. RR )
18		simpr 477	. . . . . . 7 |- ( ( N e. ZZ /\ M e. RR+ ) -> M e. RR+ )
19	18	adantr 481	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> M e. RR+ )
20		simpll 790	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> N e. ZZ )
21		modcyc 12705	. . . . . 6 |- ( ( A e. RR /\ M e. RR+ /\ N e. ZZ ) -> ( ( A + ( N x. M ) ) mod M ) = ( A mod M ) )
22	17, 19, 20, 21	syl3anc 1326	. . . . 5 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( ( A + ( N x. M ) ) mod M ) = ( A mod M ) )
23	18, 16	anim12ci 591	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( A e. RR /\ M e. RR+ ) )
24		3simpc 1060	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> ( 0 <_ A /\ A < M ) )
25	24	adantl 482	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( 0 <_ A /\ A < M ) )
26		modid 12695	. . . . . 6 |- ( ( ( A e. RR /\ M e. RR+ ) /\ ( 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
27	23, 25, 26	syl2anc 693	. . . . 5 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
28	15, 22, 27	3eqtrd 2660	. . . 4 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
29	28	ex 450	. . 3 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( ( A e. RR /\ 0 <_ A /\ A < M ) -> ( ( ( N x. M ) + A ) mod M ) = A ) )
30	5, 29	sylbid 230	. 2 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( A e. ( 0 [,) M ) -> ( ( ( N x. M ) + A ) mod M ) = A ) )
31	30	3impia 1261	1 |- ( ( N e. ZZ /\ M e. RR+ /\ A e. ( 0 [,) M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   ZZcz 11377   RR+crp 11832   [,)cico 12177   mod cmo 12668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fl 12593  df-mod 12669

This theorem is referenced by:  modmuladd  12712  addmodid  12718  mod42tp1mod8  41519