Theorem modqmuladdnn0 10585

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 2231	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
4		nn0cn 9375	. . . . . . . . . . . 12 |- ( A e. NN0 -> A e. CC )
5	4	3ad2ant1 1042	. . . . . . . . . . 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 9462	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. ZZ )
8		zq 9817	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> A e. QQ )
9	7, 8	syl 14	. . . . . . . . . . . . . . . 16 |- ( A e. NN0 -> A e. QQ )
10	9	3ad2ant1 1042	. . . . . . . . . . . . . . 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 1025	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
13		simpl3 1026	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
14	11, 12, 13	modqcld 10545	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
15		qcn 9825	. . . . . . . . . . . . 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 2292	. . . . . . . . . . . . 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 9447	. . . . . . . . . . . 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 9825	. . . . . . . . . . . . 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 8163	. . . . . . . . . 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 8472	. . . . . . . . 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 8453	. . . . . . . . . 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 9816	. . . . . . . . . . . 12 |- ( M e. QQ -> M e. RR )
33	32	3ad2ant2 1043	. . . . . . . . . . 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 8772	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M # 0 )
37	31, 22, 25, 36	divmulap3d 8968	. . . . . . . 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 6008	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
39	38	oveq1d 6015	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
40	39	eqcoms 2232	. . . . . . . . . . . 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 10552	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
44	9, 43	syl3an1 1304	. . . . . . . . . . 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 2262	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
47	46	eqeq1d 2238	. . . . . . . 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 9816	. . . . . . . . . . . 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 9390	. . . . . . . . . . . 12 |- ( A e. NN0 -> 0 <_ A )
52	51	3ad2ant1 1042	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ A )
53		simp3 1023	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 < M )
54		divge0 9016	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( A / M ) )
55	50, 52, 33, 53, 54	syl22anc 1272	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A / M ) )
56		simp2 1022	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. QQ )
57	53	gt0ne0d 8655	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M =/= 0 )
58		qdivcl 9834	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
59	10, 56, 57, 58	syl3anc 1271	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( A / M ) e. QQ )
60		0z 9453	. . . . . . . . . . 11 |- 0 e. ZZ
61		flqge 10497	. . . . . . . . . . 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 4086	. . . . . . . . 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 9455	. . . . 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 6007	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
72	71	oveq1d 6015	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
73	72	eqeq2d 2241	. . . . 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 2913	. . 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 10584	. . . . 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 1304	. . . 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 2674	. 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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   E.wrex 2509   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   CCcc 7993   RRcr 7994   0cc0 7995   + caddc 7998   x. cmul 8000   < clt 8177   <_ cle 8178   - cmin 8313   / cdiv 8815   NN0cn0 9365   ZZcz 9442   QQcq 9810   |_cfl 10483   mod cmo 10539

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-po 4386  df-iso 4387  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-n0 9366  df-z 9443  df-q 9811  df-rp 9846  df-ico 10086  df-fl 10485  df-mod 10540

This theorem is referenced by:  2lgslem3a1  15770  2lgslem3b1  15771  2lgslem3c1  15772  2lgslem3d1  15773