Theorem qnegmod 9983

Description: The negation of a number modulo a positive number is equal to the difference of the modulus and the number modulo the modulus. (Contributed by Jim Kingdon, 24-Oct-2021.)

Assertion

Ref	Expression
qnegmod	|- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Proof of Theorem qnegmod

Step	Hyp	Ref	Expression
1		qcn 9276	. . . . . 6 |- ( N e. QQ -> N e. CC )
2	1	3ad2ant2 971	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. CC )
3		qcn 9276	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 970	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> A e. CC )
5	2, 4	negsubd 7950	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N + -u A ) = ( N - A ) )
6	5	eqcomd 2105	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N - A ) = ( N + -u A ) )
7	6	oveq1d 5721	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( N - A ) mod N ) = ( ( N + -u A ) mod N ) )
8	2	mulid2d 7656	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) = N )
9	8	oveq1d 5721	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( N + -u A ) )
10	9	oveq1d 5721	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( N + -u A ) mod N ) )
11		1cnd 7654	. . . . . 6 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. CC )
12	11, 2	mulcld 7658	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) e. CC )
13		qnegcl 9278	. . . . . . 7 |- ( A e. QQ -> -u A e. QQ )
14		qcn 9276	. . . . . . 7 |- ( -u A e. QQ -> -u A e. CC )
15	13, 14	syl 14	. . . . . 6 |- ( A e. QQ -> -u A e. CC )
16	15	3ad2ant1 970	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. CC )
17	12, 16	addcomd 7784	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( -u A + ( 1 x. N ) ) )
18	17	oveq1d 5721	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( -u A + ( 1 x. N ) ) mod N ) )
19	13	3ad2ant1 970	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. QQ )
20		1zzd 8933	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. ZZ )
21		simp2 950	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. QQ )
22		simp3 951	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 0 < N )
23		modqcyc 9973	. . . 4 |- ( ( ( -u A e. QQ /\ 1 e. ZZ ) /\ ( N e. QQ /\ 0 < N ) ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
24	19, 20, 21, 22, 23	syl22anc 1185	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
25	18, 24	eqtrd 2132	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( -u A mod N ) )
26	7, 10, 25	3eqtr2rd 2139	1 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Syntax hints:   -> wi 4   /\ w3a 930   = wceq 1299   e. wcel 1448   class class class wbr 3875  (class class class)co 5706   CCcc 7498   0cc0 7500   1c1 7501   + caddc 7503   x. cmul 7505   < clt 7672   - cmin 7804   -ucneg 7805   ZZcz 8906   QQcq 9261   mod cmo 9936

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613  ax-arch 7614

This theorem depends on definitions:  df-bi 116  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210  df-div 8294  df-inn 8579  df-n0 8830  df-z 8907  df-q 9262  df-rp 9292  df-fl 9884  df-mod 9937

This theorem is referenced by:  m1modnnsub1  9984