Theorem qnegmod 10630

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 9867	. . . . . 6 |- ( N e. QQ -> N e. CC )
2	1	3ad2ant2 1045	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. CC )
3		qcn 9867	. . . . . 6 |- ( A e. QQ -> A e. CC )
4	3	3ad2ant1 1044	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> A e. CC )
5	2, 4	negsubd 8495	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N + -u A ) = ( N - A ) )
6	5	eqcomd 2237	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( N - A ) = ( N + -u A ) )
7	6	oveq1d 6032	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( N - A ) mod N ) = ( ( N + -u A ) mod N ) )
8	2	mulid2d 8197	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) = N )
9	8	oveq1d 6032	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( N + -u A ) )
10	9	oveq1d 6032	. 2 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( ( 1 x. N ) + -u A ) mod N ) = ( ( N + -u A ) mod N ) )
11		1cnd 8194	. . . . . 6 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. CC )
12	11, 2	mulcld 8199	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( 1 x. N ) e. CC )
13		qnegcl 9869	. . . . . . 7 |- ( A e. QQ -> -u A e. QQ )
14		qcn 9867	. . . . . . 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 1044	. . . . 5 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. CC )
17	12, 16	addcomd 8329	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( 1 x. N ) + -u A ) = ( -u A + ( 1 x. N ) ) )
18	17	oveq1d 6032	. . 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 1044	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> -u A e. QQ )
20		1zzd 9505	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 1 e. ZZ )
21		simp2 1024	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> N e. QQ )
22		simp3 1025	. . . 4 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> 0 < N )
23		modqcyc 10620	. . . 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 1274	. . 3 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( ( -u A + ( 1 x. N ) ) mod N ) = ( -u A mod N ) )
25	18, 24	eqtrd 2264	. 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 2271	1 |- ( ( A e. QQ /\ N e. QQ /\ 0 < N ) -> ( -u A mod N ) = ( ( N - A ) mod N ) )

Syntax hints:   -> wi 4   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   + caddc 8034   x. cmul 8036   < clt 8213   - cmin 8349   -ucneg 8350   ZZcz 9478   QQcq 9852   mod cmo 10583

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-n0 9402  df-z 9479  df-q 9853  df-rp 9888  df-fl 10529  df-mod 10584

This theorem is referenced by:  m1modnnsub1  10631  gausslemma2dlem5a  15793