Theorem modqmuladdnn0 10783

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 2240	. . . . . . . . 9 |- ( A = ( ( i x. M ) + B ) <-> ( ( i x. M ) + B ) = A )
4		nn0cn 9552	. . . . . . . . . . . 12 |- ( A e. NN0 -> A e. CC )
5	4	3ad2ant1 1049	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> A e. CC )
6	5	ad2antrr 492	. . . . . . . . . 10 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> A e. CC )
7		nn0z 9643	. . . . . . . . . . . . . . . . 17 |- ( A e. NN0 -> A e. ZZ )
8		zq 10005	. . . . . . . . . . . . . . . . 17 |- ( A e. ZZ -> A e. QQ )
9	7, 8	syl 14	. . . . . . . . . . . . . . . 16 |- ( A e. NN0 -> A e. QQ )
10	9	3ad2ant1 1049	. . . . . . . . . . . . . . 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 1032	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> M e. QQ )
13		simpl3 1033	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> 0 < M )
14	11, 12, 13	modqcld 10743	. . . . . . . . . . . . 13 |- ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) -> ( A mod M ) e. QQ )
15		qcn 10013	. . . . . . . . . . . . 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 2301	. . . . . . . . . . . . 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 9628	. . . . . . . . . . . 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 10013	. . . . . . . . . . . . 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 8336	. . . . . . . . . 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 8646	. . . . . . . . 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 8627	. . . . . . . . . 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 10004	. . . . . . . . . . . 12 |- ( M e. QQ -> M e. RR )
33	32	3ad2ant2 1050	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. RR )
34	33	ad2antrr 492	. . . . . . . . . 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 8947	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> M # 0 )
37	31, 22, 25, 36	divmulap3d 9145	. . . . . . . 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 6083	. . . . . . . . . . . . . 14 |- ( B = ( A mod M ) -> ( A - B ) = ( A - ( A mod M ) ) )
39	38	oveq1d 6090	. . . . . . . . . . . . 13 |- ( B = ( A mod M ) -> ( ( A - B ) / M ) = ( ( A - ( A mod M ) ) / M ) )
40	39	eqcoms 2241	. . . . . . . . . . . 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 10750	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
44	9, 43	syl3an1 1311	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( ( A - ( A mod M ) ) / M ) = ( |_ ` ( A / M ) ) )
45	44	ad2antrr 492	. . . . . . . . . 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 2271	. . . . . . . . 9 |- ( ( ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) /\ ( A mod M ) = B ) /\ i e. ZZ ) -> ( ( A - B ) / M ) = ( |_ ` ( A / M ) ) )
47	46	eqeq1d 2247	. . . . . . . 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 10004	. . . . . . . . . . . 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 9567	. . . . . . . . . . . 12 |- ( A e. NN0 -> 0 <_ A )
52	51	3ad2ant1 1049	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ A )
53		simp3 1030	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 < M )
54		divge0 9193	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( M e. RR /\ 0 < M ) ) -> 0 <_ ( A / M ) )
55	50, 52, 33, 53, 54	syl22anc 1279	. . . . . . . . . 10 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> 0 <_ ( A / M ) )
56		simp2 1029	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M e. QQ )
57	53	gt0ne0d 8830	. . . . . . . . . . . 12 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> M =/= 0 )
58		qdivcl 10022	. . . . . . . . . . . 12 |- ( ( A e. QQ /\ M e. QQ /\ M =/= 0 ) -> ( A / M ) e. QQ )
59	10, 56, 57, 58	syl3anc 1278	. . . . . . . . . . 11 |- ( ( A e. NN0 /\ M e. QQ /\ 0 < M ) -> ( A / M ) e. QQ )
60		0z 9634	. . . . . . . . . . 11 |- 0 e. ZZ
61		flqge 10695	. . . . . . . . . . 11 |- ( ( ( A / M ) e. QQ /\ 0 e. ZZ ) -> ( 0 <_ ( A / M ) <-> 0 <_ ( |_ ` ( A / M ) ) ) )
62	59, 60, 61	sylancl 417	. . . . . . . . . 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 4129	. . . . . . . . 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 492	. . . . . . 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 9636	. . . . 5 |- ( i e. NN0 <-> ( i e. ZZ /\ 0 <_ i ) )
70	2, 68, 69	sylanbrc 421	. . . 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 6082	. . . . . . 7 |- ( k = i -> ( k x. M ) = ( i x. M ) )
72	71	oveq1d 6090	. . . . . 6 |- ( k = i -> ( ( k x. M ) + B ) = ( ( i x. M ) + B ) )
73	72	eqeq2d 2250	. . . . 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 2935	. . 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 10782	. . . . 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 1311	. . . 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 2694	. 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 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   E.wrex 2529   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   / cdiv 8992   NN0cn0 9542   ZZcz 9623   QQcq 9998   |_cfl 10681   mod cmo 10737

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-ico 10275  df-fl 10683  df-mod 10738

This theorem is referenced by:  2lgslem3a1  16130  2lgslem3b1  16131  2lgslem3c1  16132  2lgslem3d1  16133