Theorem modqmuladdnn0 10398

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 2191	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
4		nn0cn 9215	. . . . . . . . . . . 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 9302	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. ZZ )
8		zq 9655	. . . . . . . . . . . . . . . . 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 10358	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
15		qcn 9663	. . . . . . . . . . . . 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 2252	. . . . . . . . . . . . 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 9287	. . . . . . . . . . . 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 9663	. . . . . . . . . . . . 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 8007	. . . . . . . . . 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 8316	. . . . . . . . 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 8297	. . . . . . . . . 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 9654	. . . . . . . . . . . 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 8615	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M # 0 )
37	31, 22, 25, 36	divmulap3d 8811	. . . . . . . 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 5903	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
39	38	oveq1d 5910	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
40	39	eqcoms 2192	. . . . . . . . . . . 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 10365	. . . . . . . . . . . 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 2222	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
47	46	eqeq1d 2198	. . . . . . . 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 9654	. . . . . . . . . . . 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 9230	. . . . . . . . . . . 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 8859	. . . . . . . . . . 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 8498	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M =/= 0 )
58		qdivcl 9672	. . . . . . . . . . . 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 9293	. . . . . . . . . . 11 |- 0 e. ZZ
61		flqge 10312	. . . . . . . . . . 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 4022	. . . . . . . . 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 9295	. . . . 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 5902	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
72	71	oveq1d 5910	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
73	72	eqeq2d 2201	. . . . 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 2862	. . 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 10397	. . . . 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 2633	. 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 2160   =/= wne 2360   E.wrex 2469   class class class wbr 4018   ` cfv 5235  (class class class)co 5895   CCcc 7838   RRcr 7839   0cc0 7840   + caddc 7843   x. cmul 7845   < clt 8021   <_ cle 8022   - cmin 8157   / cdiv 8658   NN0cn0 9205   ZZcz 9282   QQcq 9648   |_cfl 10298   mod cmo 10352

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1cn 7933  ax-1re 7934  ax-icn 7935  ax-addcl 7936  ax-addrcl 7937  ax-mulcl 7938  ax-mulrcl 7939  ax-addcom 7940  ax-mulcom 7941  ax-addass 7942  ax-mulass 7943  ax-distr 7944  ax-i2m1 7945  ax-0lt1 7946  ax-1rid 7947  ax-0id 7948  ax-rnegex 7949  ax-precex 7950  ax-cnre 7951  ax-pre-ltirr 7952  ax-pre-ltwlin 7953  ax-pre-lttrn 7954  ax-pre-apti 7955  ax-pre-ltadd 7956  ax-pre-mulgt0 7957  ax-pre-mulext 7958  ax-arch 7959

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-riota 5851  df-ov 5898  df-oprab 5899  df-mpo 5900  df-1st 6164  df-2nd 6165  df-pnf 8023  df-mnf 8024  df-xr 8025  df-ltxr 8026  df-le 8027  df-sub 8159  df-neg 8160  df-reap 8561  df-ap 8568  df-div 8659  df-inn 8949  df-n0 9206  df-z 9283  df-q 9649  df-rp 9683  df-ico 9923  df-fl 10300  df-mod 10353

This theorem is referenced by: (None)