Theorem muladdmodid 12137

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 9644	. . . . 5 |- ( M e. RR+ -> 0 e. RR )
2		rpxr 11309	. . . . 5 |- ( M e. RR+ -> M e. RR* )
3		elico2 11698	. . . . 5 |- ( ( 0 e. RR /\ M e. RR* ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
4	1, 2, 3	syl2anc 667	. . . 4 |- ( M e. RR+ -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
5	4	adantl 468	. . 3 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( A e. ( 0 [,) M ) <-> ( A e. RR /\ 0 <_ A /\ A < M ) ) )
6		zcn 10942	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
7		rpcn 11310	. . . . . . . . 9 |- ( M e. RR+ -> M e. CC )
8		mulcl 9623	. . . . . . . . 9 |- ( ( N e. CC /\ M e. CC ) -> ( N x. M ) e. CC )
9	6, 7, 8	syl2an 480	. . . . . . . 8 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( N x. M ) e. CC )
10	9	adantr 467	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( N x. M ) e. CC )
11		recn 9629	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
12	11	3ad2ant1 1029	. . . . . . . 8 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> A e. CC )
13	12	adantl 468	. . . . . . 7 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> A e. CC )
14	10, 13	addcomd 9835	. . . . . 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 6305	. . . . 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 1008	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> A e. RR )
17	16	adantl 468	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> A e. RR )
18		simpr 463	. . . . . . 7 |- ( ( N e. ZZ /\ M e. RR+ ) -> M e. RR+ )
19	18	adantr 467	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> M e. RR+ )
20		simpll 760	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> N e. ZZ )
21		modcyc 12132	. . . . . 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 1268	. . . . 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 571	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( A e. RR /\ M e. RR+ ) )
24		3simpc 1007	. . . . . . 7 |- ( ( A e. RR /\ 0 <_ A /\ A < M ) -> ( 0 <_ A /\ A < M ) )
25	24	adantl 468	. . . . . 6 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( 0 <_ A /\ A < M ) )
26		modid 12121	. . . . . 6 |- ( ( ( A e. RR /\ M e. RR+ ) /\ ( 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
27	23, 25, 26	syl2anc 667	. . . . 5 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( A mod M ) = A )
28	15, 22, 27	3eqtrd 2489	. . . 4 |- ( ( ( N e. ZZ /\ M e. RR+ ) /\ ( A e. RR /\ 0 <_ A /\ A < M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )
29	28	ex 436	. . 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 219	. 2 |- ( ( N e. ZZ /\ M e. RR+ ) -> ( A e. ( 0 [,) M ) -> ( ( ( N x. M ) + A ) mod M ) = A ) )
31	30	3impia 1205	1 |- ( ( N e. ZZ /\ M e. RR+ /\ A e. ( 0 [,) M ) ) -> ( ( ( N x. M ) + A ) mod M ) = A )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   class class class wbr 4402  (class class class)co 6290   CCcc 9537   RRcr 9538   0cc0 9539   + caddc 9542   x. cmul 9544   RR*cxr 9674   < clt 9675   <_ cle 9676   ZZcz 10937   RR+crp 11302   [,)cico 11637   mod cmo 12096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-ico 11641  df-fl 12028  df-mod 12097

This theorem is referenced by:  addmodid  12139