Theorem modqmuladd 9769

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 9109	. . . . . . 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 7988	. . . . . 6 |- ( ph -> M =/= 0 )
7		qdivcl 9126	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
8	3, 4, 6, 7	syl3anc 1174	. . . . 5 |- ( ph -> ( A / M ) e. QQ )
9	8	flqcld 9680	. . . 4 |- ( ph -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 5659	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 5667	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2099	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 271	. . . 4 |- ( ( ph /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14		flqpmodeq 9730	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
15	3, 4, 5, 14	syl3anc 1174	. . . . 5 |- ( ph -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
16	15	eqcomd 2093	. . . 4 |- ( ph -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
17	9, 13, 16	rspcedvd 2728	. . 3 |- ( ph -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
18		oveq2 5660	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
19	18	eqeq2d 2099	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
20	19	eqcoms 2091	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
21	20	rexbidv 2381	. . 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 155	. 2 |- ( ph -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
23		oveq1 5659	. . . . . 6 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
24	23	adantl 271	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
25		simplr 497	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> k e. ZZ )
26	4	ad2antrr 472	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> M e. QQ )
27		modqmuladd.bq	. . . . . . 7 |- ( ph -> B e. QQ )
28	27	ad2antrr 472	. . . . . 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 472	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. ( 0 [,) M ) )
31		mulqaddmodid 9767	. . . . . 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 1175	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
33	24, 32	eqtrd 2120	. . . 4 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B )
34	33	ex 113	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
35	34	rexlimdva 2489	. 2 |- ( ph -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
36	22, 35	impbid 127	1 |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   =/= wne 2255   E.wrex 2360   class class class wbr 3845   ` cfv 5015  (class class class)co 5652   0cc0 7348   + caddc 7351   x. cmul 7353   < clt 7520   / cdiv 8137   ZZcz 8748   QQcq 9102   [,)cico 9306   |_cfl 9671   mod cmo 9725

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461  ax-arch 7462

This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-po 4123  df-iso 4124  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-n0 8672  df-z 8749  df-q 9103  df-rp 9133  df-ico 9310  df-fl 9673  df-mod 9726

This theorem is referenced by:  modqmuladdim  9770