Theorem modqmuladdnn0 9370

Description: Implication of a decomposition of a nonnegative integer into a multiple of a modulus and a remainder. (Contributed by Jim Kingdon, 23-Oct-2021.)

Assertion

Ref	Expression
modqmuladdnn0	|- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )

Distinct variable groups:   A, k   B, k   k, M

Proof of Theorem modqmuladdnn0

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 108	. . . . . 6 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. ZZ )
2	1	adantr 270	. . . . 5 |- ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> i e. ZZ )
3		nn0cn 8298	. . . . . . . . . . . 12 |- ( A e. NN0 -> A e. CC )
4	3	3ad2ant1 959	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. CC )
5	4	ad2antrr 471	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> A e. CC )
6		nn0z 8371	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. ZZ )
7		zq 8711	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> A e. QQ )
8	6, 7	syl 14	. . . . . . . . . . . . . . . 16 |- ( A e. NN0 -> A e. QQ )
9	8	3ad2ant1 959	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. QQ )
10	9	adantr 270	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. QQ )
11		simpl2 942	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
12		simpl3 943	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
13	10, 11, 12	modqcld 9330	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
14		qcn 8719	. . . . . . . . . . . . 13 |- ( ( A mod M ) e. QQ -> ( A mod M ) e. CC )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. CC )
16		eleq1 2141	. . . . . . . . . . . . 13 |- ( ( A mod M ) = B -> ( ( A mod M ) e. CC <-> B e. CC ) )
17	16	adantl 271	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( ( A mod M ) e. CC <-> B e. CC ) )
18	15, 17	mpbid 145	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. CC )
19	18	adantr 270	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> B e. CC )
20		zcn 8356	. . . . . . . . . . . 12 |- ( i e. ZZ -> i e. CC )
21	20	adantl 271	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. CC )
22		qcn 8719	. . . . . . . . . . . . 13 |- ( M e. QQ -> M e. CC )
23	11, 22	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. CC )
24	23	adantr 270	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M e. CC )
25	21, 24	mulcld 7139	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( i x. M ) e. CC )
26	5, 19, 25	subadd2d 7438	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) = ( i x. M ) <-> ( ( i x. M ) + B ) = A ) )
27		eqcom 2083	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
28	26, 27	syl6rbbr 197	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) <-> ( A - B ) = ( i x. M ) ) )
29	4	adantr 270	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. CC )
30	29, 18	subcld 7419	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A - B ) e. CC )
31	30	adantr 270	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A - B ) e. CC )
32		qre 8710	. . . . . . . . . . . 12 |- ( M e. QQ -> M e. RR )
33	32	3ad2ant2 960	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. RR )
34	33	ad2antrr 471	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M e. RR )
35	12	adantr 270	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> 0 < M )
36	34, 35	gt0ap0d 7728	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M # 0 )
37	31, 21, 24, 36	divmulap3d 7911	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( ( A - B ) / M ) = i <-> ( A - B ) = ( i x. M ) ) )
38		oveq2 5540	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
39	38	oveq1d 5547	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
40	39	eqcoms 2084	. . . . . . . . . . . 12 |- ( ( A mod M ) = B -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
41	40	adantl 271	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
42	41	adantr 270	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
43		modqdiffl 9337	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
44	8, 43	syl3an1 1202	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
45	44	ad2antrr 471	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
46	42, 45	eqtrd 2113	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
47	46	eqeq1d 2089	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( ( A - B ) / M ) = i <-> ( |_ ` ( A / M ) ) = i ) )
48	28, 37, 47	3bitr2d 214	. . . . . . 7 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) <-> ( |_ ` ( A / M ) ) = i ) )
49		qre 8710	. . . . . . . . . . . 12 |- ( A e. QQ -> A e. RR )
50	9, 49	syl 14	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. RR )
51		nn0ge0 8313	. . . . . . . . . . . 12 |- ( A e. NN0 -> 0 <_ A )
52	51	3ad2ant1 959	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ A )
53		simp3 940	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 < M )
54		divge0 7951	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( A / M ) )
55	50, 52, 33, 53, 54	syl22anc 1170	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A / M ) )
56		simp2 939	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. QQ )
57	53	gt0ne0d 7613	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M =/= 0 )
58		qdivcl 8728	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
59	9, 56, 57, 58	syl3anc 1169	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( A / M ) e. QQ )
60		0z 8362	. . . . . . . . . . 11 |- 0 e. ZZ
61		flqge 9284	. . . . . . . . . . 11 |- ( ( ( A / M ) e. QQ /\ 0 e. ZZ ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
62	59, 60, 61	sylancl 404	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
63	55, 62	mpbid 145	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( |_ ` ( A / M ) ) )
64		breq2 3789	. . . . . . . . 9 |- ( ( |_ ` ( A / M ) ) = i -> ( 0 <_ ( |_ ` ( A / M ) ) <-> 0 <_ i ) )
65	63, 64	syl5ibcom 153	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( |_ ` ( A / M ) ) = i -> 0 <_ i ) )
66	65	ad2antrr 471	. . . . . . 7 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( |_ ` ( A / M ) ) = i -> 0 <_ i ) )
67	48, 66	sylbid 148	. . . . . 6 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) -> 0 <_ i ) )
68	67	imp 122	. . . . 5 |- ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> 0 <_ i )
69		elnn0z 8364	. . . . 5 |- ( i e. NN0 <-> ( i e. ZZ /\ 0 <_ i ) )
70	2, 68, 69	sylanbrc 408	. . . 4 |- ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> i e. NN0 )
71		oveq1 5539	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
72	71	oveq1d 5547	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
73	72	eqeq2d 2092	. . . . 5 |- ( k = i -> ( A = ( ( k x. M ) + B ) <-> A = ( ( i x. M ) + B ) ) )
74	73	adantl 271	. . . 4 |- ( ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) /\ k = i ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( i x. M ) + B ) ) )
75		simpr 108	. . . 4 |- ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> A = ( ( i x. M ) + B ) )
76	70, 74, 75	rspcedvd 2708	. . 3 |- ( ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> E. k e. NN0 A = ( ( k x. M ) + B ) )
77		modqmuladdim 9369	. . . . 5 |- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. i e. ZZ A = ( ( i x. M ) + B ) ) )
78	6, 77	syl3an1 1202	. . . 4 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. i e. ZZ A = ( ( i x. M ) + B ) ) )
79	78	imp 122	. . 3 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> E. i e. ZZ A = ( ( i x. M ) + B ) )
80	76, 79	r19.29a 2498	. 2 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> E. k e. NN0 A = ( ( k x. M ) + B ) )
81	80	ex 113	1 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   e. wcel 1433   =/= wne 2245   E.wrex 2349   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   + caddc 6984   x. cmul 6986   < clt 7153   <_ cle 7154   - cmin 7279   / cdiv 7760   NN0cn0 8288   ZZcz 8351   QQcq 8704   |_cfl 9272   mod cmo 9324

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-q 8705  df-rp 8735  df-ico 8917  df-fl 9274  df-mod 9325

This theorem is referenced by: (None)