Theorem modqmuladdim 10441

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 1002	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. ZZ )
3		zq 9694	. . . . . . 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 1003	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
6		simpl3 1004	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
7	4, 5, 6	modqcld 10402	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
8	1, 7	eqeltrrd 2271	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. QQ )
9		qre 9693	. . . . . 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 10406	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
12	4, 5, 6, 11	syl3anc 1249	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ ( A mod M ) )
13	12, 1	breqtrd 4056	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ B )
14		modqlt 10407	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) < M )
15	4, 5, 6, 14	syl3anc 1249	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) < M )
16	1, 15	eqbrtrrd 4054	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B < M )
17		0re 8021	. . . . . 6 |- 0 e. RR
18		qre 9693	. . . . . . 7 |- ( M e. QQ -> M e. RR )
19		rexr 8067	. . . . . . 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 10006	. . . . . 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 1182	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. ( 0 [,) M ) )
24	2, 8, 23, 5, 6	modqmuladd 10440	. . 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 980   = wceq 1364   e. wcel 2164   E.wrex 2473   class class class wbr 4030  (class class class)co 5919   RRcr 7873   0cc0 7874   + caddc 7877   x. cmul 7879   RR*cxr 8055   < clt 8056   <_ cle 8057   ZZcz 9320   QQcq 9687   [,)cico 9959   mod cmo 10396

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-q 9688  df-rp 9723  df-ico 9963  df-fl 10342  df-mod 10397

This theorem is referenced by:  modqmuladdnn0  10442  2lgsoddprmlem2  15263