Theorem modmuladd 12712

Description: Decomposition of an integer into a multiple of a modulus and a remainder. (Contributed by AV, 14-Jul-2021.)

Assertion

Ref	Expression
modmuladd	|- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( 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 modmuladd

Step	Hyp	Ref	Expression
1		zre 11381	. . . . . . . 8 |- ( A e. ZZ -> A e. RR )
2	1	adantr 481	. . . . . . 7 |- ( ( A e. ZZ /\ M e. RR+ ) -> A e. RR )
3		rpre 11839	. . . . . . . 8 |- ( M e. RR+ -> M e. RR )
4	3	adantl 482	. . . . . . 7 |- ( ( A e. ZZ /\ M e. RR+ ) -> M e. RR )
5		rpne0 11848	. . . . . . . 8 |- ( M e. RR+ -> M =/= 0 )
6	5	adantl 482	. . . . . . 7 |- ( ( A e. ZZ /\ M e. RR+ ) -> M =/= 0 )
7	2, 4, 6	redivcld 10853	. . . . . 6 |- ( ( A e. ZZ /\ M e. RR+ ) -> ( A / M ) e. RR )
8	7	flcld 12599	. . . . 5 |- ( ( A e. ZZ /\ M e. RR+ ) -> ( |_ ` ( A / M ) ) e. ZZ )
9	8	3adant2 1080	. . . 4 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( |_ ` ( A / M ) ) e. ZZ )
10		oveq1 6657	. . . . . . 7 |- ( k = ( |_ ` ( A / M ) ) -> ( k x. M ) = ( ( |_ ` ( A / M ) ) x. M ) )
11	10	oveq1d 6665	. . . . . 6 |- ( k = ( |_ ` ( A / M ) ) -> ( ( k x. M ) + ( A mod M ) ) = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
12	11	eqeq2d 2632	. . . . 5 |- ( k = ( |_ ` ( A / M ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
13	12	adantl 482	. . . 4 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k = ( |_ ` ( A / M ) ) ) -> ( A = ( ( k x. M ) + ( A mod M ) ) <-> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) ) )
14	1	anim1i 592	. . . . . . 7 |- ( ( A e. ZZ /\ M e. RR+ ) -> ( A e. RR /\ M e. RR+ ) )
15	14	3adant2 1080	. . . . . 6 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( A e. RR /\ M e. RR+ ) )
16		flpmodeq 12673	. . . . . 6 |- ( ( A e. RR /\ M e. RR+ ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
17	15, 16	syl 17	. . . . 5 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) = A )
18	17	eqcomd 2628	. . . 4 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> A = ( ( ( |_ ` ( A / M ) ) x. M ) + ( A mod M ) ) )
19	9, 13, 18	rspcedvd 3317	. . 3 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> E. k e. ZZ A = ( ( k x. M ) + ( A mod M ) ) )
20		oveq2 6658	. . . . . 6 |- ( B = ( A mod M ) -> ( ( k x. M ) + B ) = ( ( k x. M ) + ( A mod M ) ) )
21	20	eqeq2d 2632	. . . . 5 |- ( B = ( A mod M ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
22	21	eqcoms 2630	. . . 4 |- ( ( A mod M ) = B -> ( A = ( ( k x. M ) + B ) <-> A = ( ( k x. M ) + ( A mod M ) ) ) )
23	22	rexbidv 3052	. . 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 ) ) ) )
24	19, 23	syl5ibrcom 237	. 2 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. k e. ZZ A = ( ( k x. M ) + B ) ) )
25		oveq1 6657	. . . . 5 |- ( A = ( ( k x. M ) + B ) -> ( A mod M ) = ( ( ( k x. M ) + B ) mod M ) )
26		simpr 477	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> k e. ZZ )
27		simpl3 1066	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> M e. RR+ )
28		simpl2 1065	. . . . . 6 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> B e. ( 0 [,) M ) )
29		muladdmodid 12710	. . . . . 6 |- ( ( k e. ZZ /\ M e. RR+ /\ B e. ( 0 [,) M ) ) -> ( ( ( k x. M ) + B ) mod M ) = B )
30	26, 27, 28, 29	syl3anc 1326	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> ( ( ( k x. M ) + B ) mod M ) = B )
31	25, 30	sylan9eqr 2678	. . . 4 |- ( ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) /\ A = ( ( k x. M ) + B ) ) -> ( A mod M ) = B )
32	31	ex 450	. . 3 |- ( ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) /\ k e. ZZ ) -> ( A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
33	32	rexlimdva 3031	. 2 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( E. k e. ZZ A = ( ( k x. M ) + B ) -> ( A mod M ) = B ) )
34	24, 33	impbid 202	1 |- ( ( A e. ZZ /\ B e. ( 0 [,) M ) /\ M e. RR+ ) -> ( ( A mod M ) = B <-> E. k e. ZZ A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   / cdiv 10684   ZZcz 11377   RR+crp 11832   [,)cico 12177   |_cfl 12591   mod cmo 12668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fl 12593  df-mod 12669

This theorem is referenced by:  modmuladdim  12713