Theorem qnegmod 9371

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 8719	. . . . . 6 |- ( N e. QQ -> N e. CC )
2	1	3ad2ant2 960	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. CC )
3		qcn 8719	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 959	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> A e. CC )
5	2, 4	negsubd 7425	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N + -u A ) = ( N - A ) )
6	5	eqcomd 2086	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N - A ) = ( N + -u A ) )
7	6	oveq1d 5547	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( N - A ) mod N ) = ( ( N + -u A ) mod N ) )
8	2	mulid2d 7137	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) = N )
9	8	oveq1d 5547	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( N + -u A ) )
10	9	oveq1d 5547	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( N + -u A ) mod N ) )
11		1cnd 7135	. . . . . 6 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. CC )
12	11, 2	mulcld 7139	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) e. CC )
13		qnegcl 8721	. . . . . . 7 |- ( A e. QQ -> -u A e. QQ )
14		qcn 8719	. . . . . . 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 959	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. CC )
17	12, 16	addcomd 7259	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( -u A + ( 1 x. N ) ) )
18	17	oveq1d 5547	. . 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 959	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. QQ )
20		1zzd 8378	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. ZZ )
21		simp2 939	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. QQ )
22		simp3 940	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 0 < N )
23		modqcyc 9361	. . . 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 1170	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
25	18, 24	eqtrd 2113	. 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 2120	1 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Syntax hints:   -> wi 4   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   < clt 7153   - cmin 7279   -ucneg 7280   ZZcz 8351   QQcq 8704   mod cmo 9324

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-q 8705  df-rp 8735  df-fl 9274  df-mod 9325

This theorem is referenced by:  m1modnnsub1  9372