Theorem modqmuladdim 10733

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 1027	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. ZZ )
3		zq 9961	. . . . . . 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 1028	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
6		simpl3 1029	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
7	4, 5, 6	modqcld 10694	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
8	1, 7	eqeltrrd 2312	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. QQ )
9		qre 9960	. . . . . 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 10698	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
12	4, 5, 6, 11	syl3anc 1274	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ ( A mod M ) )
13	12, 1	breqtrd 4137	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ B )
14		modqlt 10699	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) < M )
15	4, 5, 6, 14	syl3anc 1274	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) < M )
16	1, 15	eqbrtrrd 4135	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B < M )
17		0re 8276	. . . . . 6 |- 0 e. RR
18		qre 9960	. . . . . . 7 |- ( M e. QQ -> M e. RR )
19		rexr 8321	. . . . . . 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 10273	. . . . . 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 1207	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. ( 0 [,) M ) )
24	2, 8, 23, 5, 6	modqmuladd 10732	. . 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 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4111  (class class class)co 6052   RRcr 8128   0cc0 8129   + caddc 8132   x. cmul 8134   RR*cxr 8309   < clt 8310   <_ cle 8311   ZZcz 9579   QQcq 9954   [,)cico 10226   mod cmo 10688

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247  ax-arch 8248

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-inn 9240  df-n0 9499  df-z 9580  df-q 9955  df-rp 9990  df-ico 10230  df-fl 10634  df-mod 10689

This theorem is referenced by:  modqmuladdnn0  10734  2lgsoddprmlem2  15996