Theorem modqmuladd 10322

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 9585	. . . . . . 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 8431	. . . . . 6 |- ( ph -> M =/= 0 )
7		qdivcl 9602	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
8	3, 4, 6, 7	syl3anc 1233	. . . . 5 |- ( ph -> ( A / M ) e. QQ )
9	8	flqcld 10233	. . . 4 |- ( ph -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 5860	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 5868	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2182	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 275	. . . 4 |- ( ( ph /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14		flqpmodeq 10283	. . . . . 6 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
15	3, 4, 5, 14	syl3anc 1233	. . . . 5 |- ( ph -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
16	15	eqcomd 2176	. . . 4 |- ( ph -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
17	9, 13, 16	rspcedvd 2840	. . 3 |- ( ph -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
18		oveq2 5861	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
19	18	eqeq2d 2182	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
20	19	eqcoms 2173	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
21	20	rexbidv 2471	. . 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 156	. 2 |- ( ph -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
23		oveq1 5860	. . . . . 6 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
24	23	adantl 275	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
25		simplr 525	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> k e. ZZ )
26	4	ad2antrr 485	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> M e. QQ )
27		modqmuladd.bq	. . . . . . 7 |- ( ph -> B e. QQ )
28	27	ad2antrr 485	. . . . . 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 485	. . . . . 6 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> B e. ( 0 [,) M ) )
31		mulqaddmodid 10320	. . . . . 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 1234	. . . . 5 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
33	24, 32	eqtrd 2203	. . . 4 |- ( ( ( ph /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B )
34	33	ex 114	. . 3 |- ( ( ph /\ k e. ZZ ) -> ( A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
35	34	rexlimdva 2587	. 2 |- ( ph -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
36	22, 35	impbid 128	1 |- ( ph -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   =/= wne 2340   E.wrex 2449   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   0cc0 7774   + caddc 7777   x. cmul 7779   < clt 7954   / cdiv 8589   ZZcz 9212   QQcq 9578   [,)cico 9847   |_cfl 10224   mod cmo 10278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579  df-rp 9611  df-ico 9851  df-fl 10226  df-mod 10279

This theorem is referenced by:  modqmuladdim  10323