Theorem divnumden 11863

Description: Calculate the reduced form of a quotient using gcd. (Contributed by Stefan O'Rear, 13-Sep-2014.)

Assertion

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

Proof of Theorem divnumden

Step	Hyp	Ref	Expression
1		simpl 108	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> A e. ZZ )
2		nnz 9066	. . . . 5 |- ( B e. NN -> B e. ZZ )
3	2	adantl 275	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
4		nnne0 8741	. . . . . . . 8 |- ( B e. NN -> B =/= 0 )
5	4	neneqd 2327	. . . . . . 7 |- ( B e. NN -> -. B = 0 )
6	5	adantl 275	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> -. B = 0 )
7	6	intnand 916	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
8		gcdn0cl 11640	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
9	1, 3, 7, 8	syl21anc 1215	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. NN )
10		gcddvds 11641	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
11	2, 10	sylan2 284	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
12		gcddiv 11696	. . . 4 |- ( ( ( A e. ZZ /\ B e. ZZ /\ ( A gcd B ) e. NN ) /\ ( ( A gcd B ) || A /\ ( A gcd B ) || B ) ) -> ( ( A gcd B ) / ( A gcd B ) ) = ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) )
13	1, 3, 9, 11, 12	syl31anc 1219	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) )
14	9	nncnd 8727	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. CC )
15	9	nnap0d 8759	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) # 0 )
16	14, 15	dividapd 8539	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
17	13, 16	eqtr3d 2172	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
18		zcn 9052	. . . 4 |- ( A e. ZZ -> A e. CC )
19	18	adantr 274	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
20		nncn 8721	. . . 4 |- ( B e. NN -> B e. CC )
21	20	adantl 275	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B e. CC )
22		simpr 109	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. NN )
23	22	nnap0d 8759	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B # 0 )
24		divcanap7 8474	. . . 4 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( ( A gcd B ) e. CC /\ ( A gcd B ) # 0 ) ) -> ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) = ( A / B ) )
25	24	eqcomd 2143	. . 3 |- ( ( A e. CC /\ ( B e. CC /\ B # 0 ) /\ ( ( A gcd B ) e. CC /\ ( A gcd B ) # 0 ) ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
26	19, 21, 23, 14, 15, 25	syl122anc 1225	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
27		znq 9409	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
28	11	simpld 111	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) || A )
29		gcdcl 11644	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
30	29	nn0zd 9164	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
31	2, 30	sylan2 284	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. ZZ )
32	9	nnne0d 8758	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) =/= 0 )
33		dvdsval2 11485	. . . . 5 |- ( ( ( A gcd B ) e. ZZ /\ ( A gcd B ) =/= 0 /\ A e. ZZ ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
34	31, 32, 1, 33	syl3anc 1216	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
35	28, 34	mpbid 146	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / ( A gcd B ) ) e. ZZ )
36	11	simprd 113	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) || B )
37		nndivdvds 11488	. . . . 5 |- ( ( B e. NN /\ ( A gcd B ) e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. NN ) )
38	22, 9, 37	syl2anc 408	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. NN ) )
39	36, 38	mpbid 146	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( B / ( A gcd B ) ) e. NN )
40		qnumdenbi 11859	. . 3 |- ( ( ( A / B ) e. QQ /\ ( A / ( A gcd B ) ) e. ZZ /\ ( B / ( A gcd B ) ) e. NN ) -> ( ( ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 /\ ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) ) <-> ( (numer ` ( A / B ) ) = ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = ( B / ( A gcd B ) ) ) ) )
41	27, 35, 39, 40	syl3anc 1216	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 /\ ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) ) <-> ( (numer ` ( A / B ) ) = ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = ( B / ( A gcd B ) ) ) ) )
42	17, 26, 41	mpbi2and 927	1 |- ( ( A e. ZZ /\ B e. NN ) -> ( (numer ` ( A / B ) ) = ( A / ( A gcd B ) ) /\ (denom ` ( A / B ) ) = ( B / ( A gcd B ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2306   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   CCcc 7611   0cc0 7613   1c1 7614   # cap 8336   / cdiv 8425   NNcn 8713   ZZcz 9047   QQcq 9404   || cdvds 11482   gcd cgcd 11624  numercnumer 11848  denomcdenom 11849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732  ax-caucvg 7733

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-sup 6864  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-2 8772  df-3 8773  df-4 8774  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fz 9784  df-fzo 9913  df-fl 10036  df-mod 10089  df-seqfrec 10212  df-exp 10286  df-cj 10607  df-re 10608  df-im 10609  df-rsqrt 10763  df-abs 10764  df-dvds 11483  df-gcd 11625  df-numer 11850  df-denom 11851

This theorem is referenced by:  divdenle  11864