Theorem modqmuladd 10032

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 9320	. . . . . . 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 8193	. . . . . 6 |- ( ph -> M =/= 0 )
7		qdivcl 9337	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
8	3, 4, 6, 7	syl3anc 1199	. . . . 5 |- ( ph -> ( A / M ) e. QQ )
9	8	flqcld 9943	. . . 4 |- ( ph -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 5735	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 5743	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2126	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 273	. . . 4 |- ( ( ph /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14		flqpmodeq 9993	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
15	3, 4, 5, 14	syl3anc 1199	. . . . 5 |- ( ph -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
16	15	eqcomd 2120	. . . 4 |- ( ph -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
17	9, 13, 16	rspcedvd 2766	. . 3 |- ( ph -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
18		oveq2 5736	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
19	18	eqeq2d 2126	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
20	19	eqcoms 2118	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
21	20	rexbidv 2412	. . 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 5735	. . . . . 6 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
24	23	adantl 273	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
25		simplr 502	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> k e. ZZ )
26	4	ad2antrr 477	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> M e. QQ )
27		modqmuladd.bq	. . . . . . 7 |- ( ph -> B e. QQ )
28	27	ad2antrr 477	. . . . . 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 477	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. ( 0 [,) M ) )
31		mulqaddmodid 10030	. . . . . 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 1200	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
33	24, 32	eqtrd 2147	. . . 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 2523	. 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 1314   e. wcel 1463   =/= wne 2282   E.wrex 2391   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   0cc0 7547   + caddc 7550   x. cmul 7552   < clt 7724   / cdiv 8345   ZZcz 8958   QQcq 9313   [,)cico 9566   |_cfl 9934   mod cmo 9988

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663  ax-arch 7664

This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-po 4178  df-iso 4179  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-n0 8882  df-z 8959  df-q 9314  df-rp 9344  df-ico 9570  df-fl 9936  df-mod 9989

This theorem is referenced by:  modqmuladdim  10033