Theorem divnumden2 29564

Description: Calculate the reduced form of a quotient using gcd. This version extends divnumden 15456 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)

Assertion

Ref	Expression
divnumden2	|- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( (numer ` ( A / B ) ) = -u ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = -u ( B / ( A gcd B ) ) ) )

Proof of Theorem divnumden2

Step	Hyp	Ref	Expression
1		zssq 11795	. . . . . . . 8 |- ZZ C_ QQ
2		simp1 1061	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> A e. ZZ )
3	1, 2	sseldi 3601	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> A e. QQ )
4		simp2 1062	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> B e. ZZ )
5	1, 4	sseldi 3601	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> B e. QQ )
6		nnne0 11053	. . . . . . . . . . . 12 |- ( -u B e. NN -> -u B =/= 0 )
7	6	3ad2ant3 1084	. . . . . . . . . . 11 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u B =/= 0 )
8		neg0 10327	. . . . . . . . . . . 12 |- -u 0 = 0
9	8	neeq2i 2859	. . . . . . . . . . 11 |- ( -u B =/= -u 0 <-> -u B =/= 0 )
10	7, 9	sylibr 224	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u B =/= -u 0 )
11	10	neneqd 2799	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -. -u B = -u 0 )
12	4	zcnd 11483	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> B e. CC )
13		0cnd 10033	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> 0 e. CC )
14	12, 13	neg11ad 10388	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( -u B = -u 0 <-> B = 0 ) )
15	11, 14	mtbid 314	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -. B = 0 )
16	15	neqned 2801	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> B =/= 0 )
17		qdivcl 11809	. . . . . . 7 |- ( ( A e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( A / B ) e. QQ )
18	3, 5, 16, 17	syl3anc 1326	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A / B ) e. QQ )
19		qnumcl 15448	. . . . . 6 |- ( ( A / B ) e. QQ -> (numer ` ( A / B ) ) e. ZZ )
20	18, 19	syl 17	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / B ) ) e. ZZ )
21	20	zcnd 11483	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / B ) ) e. CC )
22		simpl 473	. . . . . . 7 |- ( ( A e. ZZ /\ -u B e. NN ) -> A e. ZZ )
23	22	zcnd 11483	. . . . . 6 |- ( ( A e. ZZ /\ -u B e. NN ) -> A e. CC )
24	23	3adant2 1080	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> A e. CC )
25	2, 4	gcdcld 15230	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A gcd B ) e. NN0 )
26	25	nn0cnd 11353	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A gcd B ) e. CC )
27	26	negcld 10379	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A gcd B ) e. CC )
28	15	intnand 962	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
29		gcdeq0 15238	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) = 0 <-> ( A = 0 /\ B = 0 ) ) )
30	29	necon3abid 2830	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) =/= 0 <-> -. ( A = 0 /\ B = 0 ) ) )
31	30	3adant3 1081	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( ( A gcd B ) =/= 0 <-> -. ( A = 0 /\ B = 0 ) ) )
32	28, 31	mpbird 247	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A gcd B ) =/= 0 )
33	26, 32	negne0d 10390	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A gcd B ) =/= 0 )
34	24, 27, 33	divcld 10801	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A / -u ( A gcd B ) ) e. CC )
35	24, 12, 16	divneg2d 10815	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A / B ) = ( A / -u B ) )
36	35	fveq2d 6195	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` -u ( A / B ) ) = (numer ` ( A / -u B ) ) )
37		numdenneg 29563	. . . . . . 7 |- ( ( A / B ) e. QQ -> ( (numer ` -u ( A / B ) ) = -u (numer ` ( A / B ) ) /\ (denom ` -u ( A / B ) ) = (denom ` ( A / B ) ) ) )
38	37	simpld 475	. . . . . 6 |- ( ( A / B ) e. QQ -> (numer ` -u ( A / B ) ) = -u (numer ` ( A / B ) ) )
39	18, 38	syl 17	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` -u ( A / B ) ) = -u (numer ` ( A / B ) ) )
40		gcdneg 15243	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd -u B ) = ( A gcd B ) )
41	40	3adant3 1081	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A gcd -u B ) = ( A gcd B ) )
42	41	oveq2d 6666	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( A / ( A gcd -u B ) ) = ( A / ( A gcd B ) ) )
43		divnumden 15456	. . . . . . . 8 |- ( ( A e. ZZ /\ -u B e. NN ) -> ( (numer ` ( A / -u B ) ) = ( A / ( A gcd -u B ) ) /\ (denom ` ( A / -u B ) ) = ( -u B / ( A gcd -u B ) ) ) )
44	43	simpld 475	. . . . . . 7 |- ( ( A e. ZZ /\ -u B e. NN ) -> (numer ` ( A / -u B ) ) = ( A / ( A gcd -u B ) ) )
45	44	3adant2 1080	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / -u B ) ) = ( A / ( A gcd -u B ) ) )
46	24, 27, 33	divnegd 10814	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A / -u ( A gcd B ) ) = ( -u A / -u ( A gcd B ) ) )
47	24, 26, 32	div2negd 10816	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( -u A / -u ( A gcd B ) ) = ( A / ( A gcd B ) ) )
48	46, 47	eqtrd 2656	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A / -u ( A gcd B ) ) = ( A / ( A gcd B ) ) )
49	42, 45, 48	3eqtr4d 2666	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / -u B ) ) = -u ( A / -u ( A gcd B ) ) )
50	36, 39, 49	3eqtr3d 2664	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u (numer ` ( A / B ) ) = -u ( A / -u ( A gcd B ) ) )
51	21, 34, 50	neg11d 10404	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / B ) ) = ( A / -u ( A gcd B ) ) )
52	24, 26, 32	divneg2d 10815	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( A / ( A gcd B ) ) = ( A / -u ( A gcd B ) ) )
53	51, 52	eqtr4d 2659	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (numer ` ( A / B ) ) = -u ( A / ( A gcd B ) ) )
54	35	fveq2d 6195	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` -u ( A / B ) ) = (denom ` ( A / -u B ) ) )
55	37	simprd 479	. . . . 5 |- ( ( A / B ) e. QQ -> (denom ` -u ( A / B ) ) = (denom ` ( A / B ) ) )
56	18, 55	syl 17	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` -u ( A / B ) ) = (denom ` ( A / B ) ) )
57	41	oveq2d 6666	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( -u B / ( A gcd -u B ) ) = ( -u B / ( A gcd B ) ) )
58	43	simprd 479	. . . . . 6 |- ( ( A e. ZZ /\ -u B e. NN ) -> (denom ` ( A / -u B ) ) = ( -u B / ( A gcd -u B ) ) )
59	58	3adant2 1080	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` ( A / -u B ) ) = ( -u B / ( A gcd -u B ) ) )
60	12, 26, 32	divneg2d 10815	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( B / ( A gcd B ) ) = ( B / -u ( A gcd B ) ) )
61	12, 26, 32	divnegd 10814	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> -u ( B / ( A gcd B ) ) = ( -u B / ( A gcd B ) ) )
62	60, 61	eqtr3d 2658	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( B / -u ( A gcd B ) ) = ( -u B / ( A gcd B ) ) )
63	57, 59, 62	3eqtr4d 2666	. . . 4 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` ( A / -u B ) ) = ( B / -u ( A gcd B ) ) )
64	54, 56, 63	3eqtr3d 2664	. . 3 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` ( A / B ) ) = ( B / -u ( A gcd B ) ) )
65	64, 60	eqtr4d 2659	. 2 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> (denom ` ( A / B ) ) = -u ( B / ( A gcd B ) ) )
66	53, 65	jca 554	1 |- ( ( A e. ZZ /\ B e. ZZ /\ -u B e. NN ) -> ( (numer ` ( A / B ) ) = -u ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = -u ( B / ( A gcd B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   -ucneg 10267   / cdiv 10684   NNcn 11020   ZZcz 11377   QQcq 11788   gcd cgcd 15216  numercnumer 15441  denomcdenom 15442

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-1st 7168  df-2nd 7169  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-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-dvds 14984  df-gcd 15217  df-numer 15443  df-denom 15444

This theorem is referenced by:  qqhval2lem  30025