Theorem modqmuladd 10139

Description: Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)

Hypotheses

Ref	Expression
modqmuladd.a	|- ( ph -> A e. ZZ )
modqmuladd.bq	|- ( ph -> B e. QQ )
modqmuladd.b	|- ( ph -> B e. ( 0 [,) M ) )
modqmuladd.m	|- ( ph -> M e. QQ )
modqmuladd.mgt0	|- ( ph -> 0 < M )

Assertion

Ref	Expression
modqmuladd	|- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Distinct variable groups:   A, k   B, k   k, M   ph, k

Proof of Theorem modqmuladd

Step	Hyp	Ref	Expression
1		modqmuladd.a	. . . . . . 7 |- ( ph -> A e. ZZ )
2		zq 9418	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
3	1, 2	syl 14	. . . . . 6 |- ( ph -> A e. QQ )
4		modqmuladd.m	. . . . . 6 |- ( ph -> M e. QQ )
5		modqmuladd.mgt0	. . . . . . 7 |- ( ph -> 0 < M )
6	5	gt0ne0d 8274	. . . . . 6 |- ( ph -> M =/= 0 )
7		qdivcl 9435	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
8	3, 4, 6, 7	syl3anc 1216	. . . . 5 |- ( ph -> ( A / M ) e. QQ )
9	8	flqcld 10050	. . . 4 |- ( ph -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 5781	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 5789	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2151	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 275	. . . 4 |- ( ( ph /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14		flqpmodeq 10100	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
15	3, 4, 5, 14	syl3anc 1216	. . . . 5 |- ( ph -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
16	15	eqcomd 2145	. . . 4 |- ( ph -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
17	9, 13, 16	rspcedvd 2795	. . 3 |- ( ph -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
18		oveq2 5782	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
19	18	eqeq2d 2151	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
20	19	eqcoms 2142	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
21	20	rexbidv 2438	. . 3 |- ( ( A mod M ) = B -> ( E. k e. ZZ A = ( ( k x. M ) + B ) <-> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) ) )
22	17, 21	syl5ibrcom 156	. 2 |- ( ph -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
23		oveq1 5781	. . . . . 6 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
24	23	adantl 275	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
25		simplr 519	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> k e. ZZ )
26	4	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> M e. QQ )
27		modqmuladd.bq	. . . . . . 7 |- ( ph -> B e. QQ )
28	27	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. QQ )
29		modqmuladd.b	. . . . . . 7 |- ( ph -> B e. ( 0 [,) M ) )
30	29	ad2antrr 479	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. ( 0 [,) M ) )
31		mulqaddmodid 10137	. . . . . 6 |- ( ( ( k e. ZZ /\ M e. QQ ) /\ ( B e. QQ /\ B e. ( 0 [,) M ) ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
32	25, 26, 28, 30, 31	syl22anc 1217	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
33	24, 32	eqtrd 2172	. . . 4 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B )
34	33	ex 114	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
35	34	rexlimdva 2549	. 2 |- ( ph -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
36	22, 35	impbid 128	1 |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   =/= wne 2308   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   0cc0 7620   + caddc 7623   x. cmul 7625   < clt 7800   / cdiv 8432   ZZcz 9054   QQcq 9411   [,)cico 9673   |_cfl 10041   mod cmo 10095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738  ax-arch 7739

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-q 9412  df-rp 9442  df-ico 9677  df-fl 10043  df-mod 10096

This theorem is referenced by:  modqmuladdim  10140