Theorem modqmuladdim 10584

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

Assertion

Ref	Expression
modqmuladdim	|- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

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

Proof of Theorem modqmuladdim

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) = B )
2		simpl1 1024	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. ZZ )
3		zq 9817	. . . . . . 7 |- ( A e. ZZ -> A e. QQ )
4	2, 3	syl 14	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. QQ )
5		simpl2 1025	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
6		simpl3 1026	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
7	4, 5, 6	modqcld 10545	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
8	1, 7	eqeltrrd 2307	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. QQ )
9		qre 9816	. . . . . 6 |- ( B e. QQ -> B e. RR )
10	8, 9	syl 14	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. RR )
11		modqge0 10549	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
12	4, 5, 6, 11	syl3anc 1271	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ ( A mod M ) )
13	12, 1	breqtrd 4108	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ B )
14		modqlt 10550	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) < M )
15	4, 5, 6, 14	syl3anc 1271	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) < M )
16	1, 15	eqbrtrrd 4106	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B < M )
17		0re 8142	. . . . . 6 |- 0 e. RR
18		qre 9816	. . . . . . 7 |- ( M e. QQ -> M e. RR )
19		rexr 8188	. . . . . . 7 |- ( M e. RR -> M e. RR* )
20	5, 18, 19	3syl 17	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. RR* )
21		elico2 10129	. . . . . 6 |- ( ( 0 e. RR /\ M e. RR* ) -> ( B e. ( 0 [,) M ) <-> ( B e. RR /\ 0 <_ B /\ B < M ) ) )
22	17, 20, 21	sylancr 414	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( B e. ( 0 [,) M ) <-> ( B e. RR /\ 0 <_ B /\ B < M ) ) )
23	10, 13, 16, 22	mpbir3and 1204	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. ( 0 [,) M ) )
24	2, 8, 23, 5, 6	modqmuladd 10583	. . 3 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
25	1, 24	mpbid 147	. 2 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> E. k e. ZZ A = ( ( k x. M ) + B ) )
26	25	ex 115	1 |- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   RRcr 7994   0cc0 7995   + caddc 7998   x. cmul 8000   RR*cxr 8176   < clt 8177   <_ cle 8178   ZZcz 9442   QQcq 9810   [,)cico 10082   mod cmo 10539

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-ico 10086  df-fl 10485  df-mod 10540

This theorem is referenced by:  modqmuladdnn0  10585  2lgsoddprmlem2  15779