Theorem quoremz 12654

Description: Quotient and remainder of an integer divided by a positive integer. TODO - is this really needed for anything? Should we use mod to simplify it? Remark (AV): This is a special case of divalg 15126. (Contributed by NM, 14-Aug-2008.)

Hypotheses

Ref	Expression
quorem.1	|- Q = ( |_ ` ( A / B ) )
quorem.2	|- R = ( A - ( B x. Q ) )

Assertion

Ref	Expression
quoremz	|- ( ( A e. ZZ /\ B e. NN ) -> ( ( Q e. ZZ /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) )

Proof of Theorem quoremz

Step	Hyp	Ref	Expression
1		quorem.1	. . 3 |- Q = ( |_ ` ( A / B ) )
2		zre 11381	. . . . . 6 |- ( A e. ZZ -> A e. RR )
3	2	adantr 481	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> A e. RR )
4		nnre 11027	. . . . . 6 |- ( B e. NN -> B e. RR )
5	4	adantl 482	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> B e. RR )
6		nnne0 11053	. . . . . 6 |- ( B e. NN -> B =/= 0 )
7	6	adantl 482	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> B =/= 0 )
8	3, 5, 7	redivcld 10853	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. RR )
9	8	flcld 12599	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( |_ ` ( A / B ) ) e. ZZ )
10	1, 9	syl5eqel 2705	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> Q e. ZZ )
11		quorem.2	. . 3 |- R = ( A - ( B x. Q ) )
12	10	zcnd 11483	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> Q e. CC )
13		nncn 11028	. . . . . . . 8 |- ( B e. NN -> B e. CC )
14	13	adantl 482	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> B e. CC )
15	12, 14, 7	divcan3d 10806	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( B x. Q ) / B ) = Q )
16		flle 12600	. . . . . . . 8 |- ( ( A / B ) e. RR -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
17	8, 16	syl 17	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> ( |_ ` ( A / B ) ) <_ ( A / B ) )
18	1, 17	syl5eqbr 4688	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> Q <_ ( A / B ) )
19	15, 18	eqbrtrd 4675	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( B x. Q ) / B ) <_ ( A / B ) )
20		nnz 11399	. . . . . . . . 9 |- ( B e. NN -> B e. ZZ )
21	20	adantl 482	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
22	21, 10	zmulcld 11488	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> ( B x. Q ) e. ZZ )
23	22	zred 11482	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> ( B x. Q ) e. RR )
24		nngt0 11049	. . . . . . 7 |- ( B e. NN -> 0 < B )
25	24	adantl 482	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> 0 < B )
26		lediv1 10888	. . . . . 6 |- ( ( ( B x. Q ) e. RR /\ A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( B x. Q ) <_ A <-> ( ( B x. Q ) / B ) <_ ( A / B ) ) )
27	23, 3, 5, 25, 26	syl112anc 1330	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( B x. Q ) <_ A <-> ( ( B x. Q ) / B ) <_ ( A / B ) ) )
28	19, 27	mpbird 247	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( B x. Q ) <_ A )
29		simpl 473	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> A e. ZZ )
30		znn0sub 11424	. . . . 5 |- ( ( ( B x. Q ) e. ZZ /\ A e. ZZ ) -> ( ( B x. Q ) <_ A <-> ( A - ( B x. Q ) ) e. NN0 ) )
31	22, 29, 30	syl2anc 693	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( B x. Q ) <_ A <-> ( A - ( B x. Q ) ) e. NN0 ) )
32	28, 31	mpbid 222	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( A - ( B x. Q ) ) e. NN0 )
33	11, 32	syl5eqel 2705	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> R e. NN0 )
34	1	oveq2i 6661	. . . . . 6 |- ( ( A / B ) - Q ) = ( ( A / B ) - ( |_ ` ( A / B ) ) )
35		fraclt1 12603	. . . . . . 7 |- ( ( A / B ) e. RR -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
36	8, 35	syl 17	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / B ) - ( |_ ` ( A / B ) ) ) < 1 )
37	34, 36	syl5eqbr 4688	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / B ) - Q ) < 1 )
38	11	oveq1i 6660	. . . . . 6 |- ( R / B ) = ( ( A - ( B x. Q ) ) / B )
39		zcn 11382	. . . . . . . . 9 |- ( A e. ZZ -> A e. CC )
40	39	adantr 481	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
41	22	zcnd 11483	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. NN ) -> ( B x. Q ) e. CC )
42	13, 6	jca 554	. . . . . . . . 9 |- ( B e. NN -> ( B e. CC /\ B =/= 0 ) )
43	42	adantl 482	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. NN ) -> ( B e. CC /\ B =/= 0 ) )
44		divsubdir 10721	. . . . . . . 8 |- ( ( A e. CC /\ ( B x. Q ) e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( ( A - ( B x. Q ) ) / B ) = ( ( A / B ) - ( ( B x. Q ) / B ) ) )
45	40, 41, 43, 44	syl3anc 1326	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A - ( B x. Q ) ) / B ) = ( ( A / B ) - ( ( B x. Q ) / B ) ) )
46	15	oveq2d 6666	. . . . . . 7 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / B ) - ( ( B x. Q ) / B ) ) = ( ( A / B ) - Q ) )
47	45, 46	eqtrd 2656	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A - ( B x. Q ) ) / B ) = ( ( A / B ) - Q ) )
48	38, 47	syl5eq 2668	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( R / B ) = ( ( A / B ) - Q ) )
49	13, 6	dividd 10799	. . . . . 6 |- ( B e. NN -> ( B / B ) = 1 )
50	49	adantl 482	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( B / B ) = 1 )
51	37, 48, 50	3brtr4d 4685	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( R / B ) < ( B / B ) )
52	33	nn0red 11352	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> R e. RR )
53		ltdiv1 10887	. . . . 5 |- ( ( R e. RR /\ B e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( R < B <-> ( R / B ) < ( B / B ) ) )
54	52, 5, 5, 25, 53	syl112anc 1330	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( R < B <-> ( R / B ) < ( B / B ) ) )
55	51, 54	mpbird 247	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> R < B )
56	11	oveq2i 6661	. . . 4 |- ( ( B x. Q ) + R ) = ( ( B x. Q ) + ( A - ( B x. Q ) ) )
57	41, 40	pncan3d 10395	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( B x. Q ) + ( A - ( B x. Q ) ) ) = A )
58	56, 57	syl5req 2669	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A = ( ( B x. Q ) + R ) )
59	55, 58	jca 554	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( R < B /\ A = ( ( B x. Q ) + R ) ) )
60	10, 33, 59	jca31 557	1 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( Q e. ZZ /\ R e. NN0 ) /\ ( R < B /\ A = ( ( B x. Q ) + R ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   |_cfl 12591

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-fl 12593

This theorem is referenced by:  quoremnn0  12655