Theorem qnegmod 10399

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 9663	. . . . . 6 |- ( N e. QQ -> N e. CC )
2	1	3ad2ant2 1021	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. CC )
3		qcn 9663	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 1020	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> A e. CC )
5	2, 4	negsubd 8303	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N + -u A ) = ( N - A ) )
6	5	eqcomd 2195	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N - A ) = ( N + -u A ) )
7	6	oveq1d 5910	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( N - A ) mod N ) = ( ( N + -u A ) mod N ) )
8	2	mulid2d 8005	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) = N )
9	8	oveq1d 5910	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( N + -u A ) )
10	9	oveq1d 5910	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( N + -u A ) mod N ) )
11		1cnd 8002	. . . . . 6 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. CC )
12	11, 2	mulcld 8007	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) e. CC )
13		qnegcl 9665	. . . . . . 7 |- ( A e. QQ -> -u A e. QQ )
14		qcn 9663	. . . . . . 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 1020	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. CC )
17	12, 16	addcomd 8137	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( -u A + ( 1 x. N ) ) )
18	17	oveq1d 5910	. . 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 1020	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. QQ )
20		1zzd 9309	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. ZZ )
21		simp2 1000	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. QQ )
22		simp3 1001	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 0 < N )
23		modqcyc 10389	. . . 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 1250	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
25	18, 24	eqtrd 2222	. 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 2229	1 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4018  (class class class)co 5895   CCcc 7838   0cc0 7840   1c1 7841   + caddc 7843   x. cmul 7845   < clt 8021   - cmin 8157   -ucneg 8158   ZZcz 9282   QQcq 9648   mod cmo 10352

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-fl 10300  df-mod 10353

This theorem is referenced by:  m1modnnsub1  10400