Theorem modmuladdnn0 12714

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

Assertion

Ref	Expression
modmuladdnn0	|- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( 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 modmuladdnn0

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 477	. . . . . 6 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. ZZ )
2	1	adantr 481	. . . . 5 |- ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> i e. ZZ )
3		nn0cn 11302	. . . . . . . . . . . 12 |- ( A e. NN0 -> A e. CC )
4	3	adantr 481	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. RR+ ) -> A e. CC )
5	4	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> A e. CC )
6		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( A e. NN0 -> A e. RR )
7		modcl 12672	. . . . . . . . . . . . . . 15 |- ( ( A e. RR /\ M e. RR+ ) -> ( A mod M ) e. RR )
8	6, 7	sylan 488	. . . . . . . . . . . . . 14 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( A mod M ) e. RR )
9	8	recnd 10068	. . . . . . . . . . . . 13 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( A mod M ) e. CC )
10	9	adantr 481	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> ( A mod M ) e. CC )
11		eleq1 2689	. . . . . . . . . . . . 13 |- ( ( A mod M ) = B -> ( ( A mod M ) e. CC <-> B e. CC ) )
12	11	adantl 482	. . . . . . . . . . . 12 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> ( ( A mod M ) e. CC <-> B e. CC ) )
13	10, 12	mpbid 222	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> B e. CC )
14	13	adantr 481	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> B e. CC )
15		zcn 11382	. . . . . . . . . . . 12 |- ( i e. ZZ -> i e. CC )
16	15	adantl 482	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> i e. CC )
17		rpcn 11841	. . . . . . . . . . . . 13 |- ( M e. RR+ -> M e. CC )
18	17	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. RR+ ) -> M e. CC )
19	18	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M e. CC )
20	16, 19	mulcld 10060	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( i x. M ) e. CC )
21	5, 14, 20	subadd2d 10411	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) = ( i x. M ) <-> ( ( i x. M ) + B ) = A ) )
22		eqcom 2629	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
23	21, 22	syl6rbbr 279	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) <-> ( A - B ) = ( i x. M ) ) )
24	3	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> A e. CC )
25	24, 13	subcld 10392	. . . . . . . . . 10 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> ( A - B ) e. CC )
26	25	adantr 481	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A - B ) e. CC )
27		rpcnne0 11850	. . . . . . . . . . 11 |- ( M e. RR+ -> ( M e. CC /\ M =/= 0 ) )
28	27	adantl 482	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( M e. CC /\ M =/= 0 ) )
29	28	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( M e. CC /\ M =/= 0 ) )
30		divmul3 10690	. . . . . . . . 9 |- ( ( ( A - B ) e. CC /\ i e. CC /\ ( M e. CC /\ M =/= 0 ) ) -> ( ( ( A - B ) / M ) = i <-> ( A - B ) = ( i x. M ) ) )
31	26, 16, 29, 30	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( ( A - B ) / M ) = i <-> ( A - B ) = ( i x. M ) ) )
32		oveq2 6658	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
33	32	oveq1d 6665	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
34	33	eqcoms 2630	. . . . . . . . . . . 12 |- ( ( A mod M ) = B -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
35	34	adantl 482	. . . . . . . . . . 11 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
36	35	adantr 481	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
37		moddiffl 12681	. . . . . . . . . . . 12 |- ( ( A e. RR /\ M e. RR+ ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
38	6, 37	sylan 488	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
39	38	ad2antrr 762	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
40	36, 39	eqtrd 2656	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
41	40	eqeq1d 2624	. . . . . . . 8 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( ( A - B ) / M ) = i <-> ( |_ ` ( A / M ) ) = i ) )
42	23, 31, 41	3bitr2d 296	. . . . . . 7 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) <-> ( |_ ` ( A / M ) ) = i ) )
43		nn0ge0 11318	. . . . . . . . . . . 12 |- ( A e. NN0 -> 0 <_ A )
44	6, 43	jca 554	. . . . . . . . . . 11 |- ( A e. NN0 -> ( A e. RR /\ 0 <_ A ) )
45		rpregt0 11846	. . . . . . . . . . 11 |- ( M e. RR+ -> ( M e. RR /\ 0 < M ) )
46		divge0 10892	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( A / M ) )
47	44, 45, 46	syl2an 494	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. RR+ ) -> 0 <_ ( A / M ) )
48	6	adantr 481	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. RR+ ) -> A e. RR )
49		rpre 11839	. . . . . . . . . . . . 13 |- ( M e. RR+ -> M e. RR )
50	49	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. RR+ ) -> M e. RR )
51		rpne0 11848	. . . . . . . . . . . . 13 |- ( M e. RR+ -> M =/= 0 )
52	51	adantl 482	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. RR+ ) -> M =/= 0 )
53	48, 50, 52	redivcld 10853	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( A / M ) e. RR )
54		0z 11388	. . . . . . . . . . 11 |- 0 e. ZZ
55		flge 12606	. . . . . . . . . . 11 |- ( ( ( A / M ) e. RR /\ 0 e. ZZ ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
56	53, 54, 55	sylancl 694	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
57	47, 56	mpbid 222	. . . . . . . . 9 |- ( ( A e. NN0 /\ M e. RR+ ) -> 0 <_ ( |_ ` ( A / M ) ) )
58		breq2 4657	. . . . . . . . 9 |- ( ( |_ ` ( A / M ) ) = i -> ( 0 <_ ( |_ ` ( A / M ) ) <-> 0 <_ i ) )
59	57, 58	syl5ibcom 235	. . . . . . . 8 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( |_ ` ( A / M ) ) = i -> 0 <_ i ) )
60	59	ad2antrr 762	. . . . . . 7 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( |_ ` ( A / M ) ) = i -> 0 <_ i ) )
61	42, 60	sylbid 230	. . . . . 6 |- ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( A = ( ( i x. M ) + B ) -> 0 <_ i ) )
62	61	imp 445	. . . . 5 |- ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> 0 <_ i )
63		elnn0z 11390	. . . . 5 |- ( i e. NN0 <-> ( i e. ZZ /\ 0 <_ i ) )
64	2, 62, 63	sylanbrc 698	. . . 4 |- ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> i e. NN0 )
65		oveq1 6657	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
66	65	oveq1d 6665	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
67	66	eqeq2d 2632	. . . . 5 |- ( k = i -> ( A = ( ( k x. M ) + B ) <-> A = ( ( i x. M ) + B ) ) )
68	67	adantl 482	. . . 4 |- ( ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) /\ k = i ) -> ( A = ( ( k x. M ) + B ) <-> A = ( ( i x. M ) + B ) ) )
69		simpr 477	. . . 4 |- ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> A = ( ( i x. M ) + B ) )
70	64, 68, 69	rspcedvd 3317	. . 3 |- ( ( ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) /\ i e. ZZ ) /\ A = ( ( i x. M ) + B ) ) -> E. k e. NN0 A = ( ( k x. M ) + B ) )
71		nn0z 11400	. . . . 5 |- ( A e. NN0 -> A e. ZZ )
72		modmuladdim 12713	. . . . 5 |- ( ( A e. ZZ /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. i e. ZZ A = ( ( i x. M ) + B ) ) )
73	71, 72	sylan 488	. . . 4 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. i e. ZZ A = ( ( i x. M ) + B ) ) )
74	73	imp 445	. . 3 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> E. i e. ZZ A = ( ( i x. M ) + B ) )
75	70, 74	r19.29a 3078	. 2 |- ( ( ( A e. NN0 /\ M e. RR+ ) /\ ( A mod M ) = B ) -> E. k e. NN0 A = ( ( k x. M ) + B ) )
76	75	ex 450	1 |- ( ( A e. NN0 /\ M e. RR+ ) -> ( ( A mod M ) = B -> E. k e. NN0 A = ( ( k x. M ) + B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NN0cn0 11292   ZZcz 11377   RR+crp 11832   |_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:  2lgslem3a1  25125  2lgslem3b1  25126  2lgslem3c1  25127  2lgslem3d1  25128