Theorem modqmuladd 10509

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 9746	. . . . . . 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 8584	. . . . . 6 |- ( ph -> M =/= 0 )
7		qdivcl 9763	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
8	3, 4, 6, 7	syl3anc 1249	. . . . 5 |- ( ph -> ( A / M ) e. QQ )
9	8	flqcld 10418	. . . 4 |- ( ph -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 5950	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 5958	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2216	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 277	. . . 4 |- ( ( ph /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14		flqpmodeq 10470	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
15	3, 4, 5, 14	syl3anc 1249	. . . . 5 |- ( ph -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
16	15	eqcomd 2210	. . . 4 |- ( ph -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
17	9, 13, 16	rspcedvd 2882	. . 3 |- ( ph -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
18		oveq2 5951	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
19	18	eqeq2d 2216	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
20	19	eqcoms 2207	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
21	20	rexbidv 2506	. . 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 157	. 2 |- ( ph -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
23		oveq1 5950	. . . . . 6 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
24	23	adantl 277	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
25		simplr 528	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> k e. ZZ )
26	4	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> M e. QQ )
27		modqmuladd.bq	. . . . . . 7 |- ( ph -> B e. QQ )
28	27	ad2antrr 488	. . . . . 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 488	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. ( 0 [,) M ) )
31		mulqaddmodid 10507	. . . . . 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 1250	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
33	24, 32	eqtrd 2237	. . . 4 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B )
34	33	ex 115	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
35	34	rexlimdva 2622	. 2 |- ( ph -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
36	22, 35	impbid 129	1 |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1372   e. wcel 2175   =/= wne 2375   E.wrex 2484   class class class wbr 4043   ` cfv 5270  (class class class)co 5943   0cc0 7924   + caddc 7927   x. cmul 7929   < clt 8106   / cdiv 8744   ZZcz 9371   QQcq 9739   [,)cico 10011   |_cfl 10409   mod cmo 10465

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-po 4342  df-iso 4343  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-n0 9295  df-z 9372  df-q 9740  df-rp 9775  df-ico 10015  df-fl 10411  df-mod 10466

This theorem is referenced by:  modqmuladdim  10510