Theorem qnegmod 10731

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 9966	. . . . . 6 |- ( N e. QQ -> N e. CC )
2	1	3ad2ant2 1046	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. CC )
3		qcn 9966	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 1045	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> A e. CC )
5	2, 4	negsubd 8590	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N + -u A ) = ( N - A ) )
6	5	eqcomd 2238	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N - A ) = ( N + -u A ) )
7	6	oveq1d 6065	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( N - A ) mod N ) = ( ( N + -u A ) mod N ) )
8	2	mullidd 8292	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) = N )
9	8	oveq1d 6065	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( N + -u A ) )
10	9	oveq1d 6065	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( N + -u A ) mod N ) )
11		1cnd 8290	. . . . . 6 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. CC )
12	11, 2	mulcld 8294	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) e. CC )
13		qnegcl 9968	. . . . . . 7 |- ( A e. QQ -> -u A e. QQ )
14		qcn 9966	. . . . . . 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 1045	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. CC )
17	12, 16	addcomd 8424	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( -u A + ( 1 x. N ) ) )
18	17	oveq1d 6065	. . 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 1045	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. QQ )
20		1zzd 9604	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. ZZ )
21		simp2 1025	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. QQ )
22		simp3 1026	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 0 < N )
23		modqcyc 10721	. . . 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 1275	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
25	18, 24	eqtrd 2265	. 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 2272	1 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   - cmin 8444   -ucneg 8445   ZZcz 9577   QQcq 9951   mod cmo 10684

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-n0 9497  df-z 9578  df-q 9952  df-rp 9987  df-fl 10630  df-mod 10685

This theorem is referenced by:  m1modnnsub1  10732  gausslemma2dlem5a  15938