Theorem modqmuladdim 10782

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 1031	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. ZZ )
3		zq 10005	. . . . . . 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 1032	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
6		simpl3 1033	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
7	4, 5, 6	modqcld 10743	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
8	1, 7	eqeltrrd 2316	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. QQ )
9		qre 10004	. . . . . 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 10747	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A mod M ) )
12	4, 5, 6, 11	syl3anc 1278	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ ( A mod M ) )
13	12, 1	breqtrd 4151	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 <_ B )
14		modqlt 10748	. . . . . . 7 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( A mod M ) < M )
15	4, 5, 6, 14	syl3anc 1278	. . . . . 6 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) < M )
16	1, 15	eqbrtrrd 4149	. . . . 5 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B < M )
17		0re 8316	. . . . . 6 |- 0 e. RR
18		qre 10004	. . . . . . 7 |- ( M e. QQ -> M e. RR )
19		rexr 8361	. . . . . . 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 10318	. . . . . 6 |- ( ( 0 e. RR /\ M e. RR* ) -> ( B e. ( 0 [,) M ) <-> ( B e. RR /\ 0 <_ B /\ B < M ) ) )
22	17, 20, 21	sylancr 418	. . . . 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 1211	. . . 4 |- ( ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. ( 0 [,) M ) )
24	2, 8, 23, 5, 6	modqmuladd 10781	. . 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   + caddc 8172   x. cmul 8174   RR*cxr 8349   < clt 8350   <_ cle 8351   ZZcz 9623   QQcq 9998   [,)cico 10271   mod cmo 10737

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-ico 10275  df-fl 10683  df-mod 10738

This theorem is referenced by:  modqmuladdnn0  10783  2lgsoddprmlem2  16139