Theorem modqmuladdnn0 10462

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 110	. . . . . 6 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. ZZ )
2	1	adantr 276	. . . . 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		eqcom 2198	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
4		nn0cn 9261	. . . . . . . . . . . 12 |- ( A e. NN0 -> A e. CC )
5	4	3ad2ant1 1020	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. CC )
6	5	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> A e. CC )
7		nn0z 9348	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. ZZ )
8		zq 9702	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> A e. QQ )
9	7, 8	syl 14	. . . . . . . . . . . . . . . 16 |- ( A e. NN0 -> A e. QQ )
10	9	3ad2ant1 1020	. . . . . . . . . . . . . . 15 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. QQ )
11	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. QQ )
12		simpl2 1003	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
13		simpl3 1004	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
14	11, 12, 13	modqcld 10422	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
15		qcn 9710	. . . . . . . . . . . . 13 |- ( ( A mod M ) e. QQ -> ( A mod M ) e. CC )
16	14, 15	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. CC )
17		eleq1 2259	. . . . . . . . . . . . 13 |- ( ( A mod M ) = B -> ( ( A mod M ) e. CC <-> B e. CC ) )
18	17	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( ( A mod M ) e. CC <-> B e. CC ) )
19	16, 18	mpbid 147	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> B e. CC )
20	19	adantr 276	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> B e. CC )
21		zcn 9333	. . . . . . . . . . . 12 |- ( i e. ZZ -> i e. CC )
22	21	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. CC )
23		qcn 9710	. . . . . . . . . . . . 13 |- ( M e. QQ -> M e. CC )
24	12, 23	syl 14	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. CC )
25	24	adantr 276	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M e. CC )
26	22, 25	mulcld 8049	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( i x. M ) e. CC )
27	6, 20, 26	subadd2d 8358	. . . . . . . . 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 ) )
28	3, 27	bitr4id 199	. . . . . . . 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	5	adantr 276	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> A e. CC )
30	29, 19	subcld 8339	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A - B ) e. CC )
31	30	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A - B ) e. CC )
32		qre 9701	. . . . . . . . . . . 12 |- ( M e. QQ -> M e. RR )
33	32	3ad2ant2 1021	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. RR )
34	33	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M e. RR )
35	13	adantr 276	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> 0 < M )
36	34, 35	gt0ap0d 8658	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M # 0 )
37	31, 22, 25, 36	divmulap3d 8854	. . . . . . . 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 5931	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
39	38	oveq1d 5938	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
40	39	eqcoms 2199	. . . . . . . . . . . 12 |- ( ( A mod M ) = B -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
41	40	adantl 277	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
42	41	adantr 276	. . . . . . . . . 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 10429	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
44	9, 43	syl3an1 1282	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
45	44	ad2antrr 488	. . . . . . . . . 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 2229	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
47	46	eqeq1d 2205	. . . . . . . 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 216	. . . . . . 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 9701	. . . . . . . . . . . 12 |- ( A e. QQ -> A e. RR )
50	10, 49	syl 14	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. RR )
51		nn0ge0 9276	. . . . . . . . . . . 12 |- ( A e. NN0 -> 0 <_ A )
52	51	3ad2ant1 1020	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ A )
53		simp3 1001	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 < M )
54		divge0 8902	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( A / M ) )
55	50, 52, 33, 53, 54	syl22anc 1250	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A / M ) )
56		simp2 1000	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. QQ )
57	53	gt0ne0d 8541	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M =/= 0 )
58		qdivcl 9719	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
59	10, 56, 57, 58	syl3anc 1249	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( A / M ) e. QQ )
60		0z 9339	. . . . . . . . . . 11 |- 0 e. ZZ
61		flqge 10374	. . . . . . . . . . 11 |- ( ( ( A / M ) e. QQ /\ 0 e. ZZ ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
62	59, 60, 61	sylancl 413	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
63	55, 62	mpbid 147	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( |_ ` ( A / M ) ) )
64		breq2 4038	. . . . . . . . 9 |- ( ( |_ ` ( A / M ) ) = i -> ( 0 <_ ( |_ ` ( A / M ) ) <-> 0 <_ i ) )
65	63, 64	syl5ibcom 155	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( |_ ` ( A / M ) ) = i -> 0 <_ i ) )
66	65	ad2antrr 488	. . . . . . 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 150	. . . . . 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 124	. . . . 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 9341	. . . . 5 |- ( i e. NN0 <-> ( i e. ZZ /\ 0 <_ i ) )
70	2, 68, 69	sylanbrc 417	. . . 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 5930	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
72	71	oveq1d 5938	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
73	72	eqeq2d 2208	. . . . 5 |- ( k = i -> ( A = ( ( k x. M ) + B ) <-> A = ( ( i x. M ) + B ) ) )
74	73	adantl 277	. . . 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 110	. . . 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 2874	. . 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 10461	. . . . 5 |- ( ( A e. ZZ /\ M e. QQ /\ 0 < M ) -> ( ( A mod M ) = B -> E. i e. ZZ A = ( ( i x. M ) + B ) ) )
78	7, 77	syl3an1 1282	. . . 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 124	. . 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 2640	. 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 115	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   =/= wne 2367   E.wrex 2476   class class class wbr 4034   ` cfv 5259  (class class class)co 5923   CCcc 7879   RRcr 7880   0cc0 7881   + caddc 7884   x. cmul 7886   < clt 8063   <_ cle 8064   - cmin 8199   / cdiv 8701   NN0cn0 9251   ZZcz 9328   QQcq 9695   |_cfl 10360   mod cmo 10416

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999  ax-arch 8000

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993  df-n0 9252  df-z 9329  df-q 9696  df-rp 9731  df-ico 9971  df-fl 10362  df-mod 10417

This theorem is referenced by:  2lgslem3a1  15348  2lgslem3b1  15349  2lgslem3c1  15350  2lgslem3d1  15351