Theorem divnumden 12209

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 109	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> A e. ZZ )
2		nnz 9285	. . . . 5 |- ( B e. NN -> B e. ZZ )
3	2	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
4		nnne0 8960	. . . . . . . 8 |- ( B e. NN -> B =/= 0 )
5	4	neneqd 2378	. . . . . . 7 |- ( B e. NN -> -. B = 0 )
6	5	adantl 277	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> -. B = 0 )
7	6	intnand 932	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
8		gcdn0cl 11976	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
9	1, 3, 7, 8	syl21anc 1247	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. NN )
10		gcddvds 11977	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
11	2, 10	sylan2 286	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || A /\ ( A gcd B ) || B ) )
12		gcddiv 12033	. . . 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 1251	. . 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 8946	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. CC )
15	9	nnap0d 8978	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) # 0 )
16	14, 15	dividapd 8756	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
17	13, 16	eqtr3d 2222	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
18		zcn 9271	. . . 4 |- ( A e. ZZ -> A e. CC )
19	18	adantr 276	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
20		nncn 8940	. . . 4 |- ( B e. NN -> B e. CC )
21	20	adantl 277	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B e. CC )
22		simpr 110	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. NN )
23	22	nnap0d 8978	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B # 0 )
24		divcanap7 8691	. . . 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 2193	. . 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 1257	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
27		znq 9637	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) e. QQ )
28	11	simpld 112	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) || A )
29		gcdcl 11980	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
30	29	nn0zd 9386	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. ZZ )
31	2, 30	sylan2 286	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. ZZ )
32	9	nnne0d 8977	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) =/= 0 )
33		dvdsval2 11810	. . . . 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 1248	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || A <-> ( A / ( A gcd B ) ) e. ZZ ) )
35	28, 34	mpbid 147	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / ( A gcd B ) ) e. ZZ )
36	11	simprd 114	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) || B )
37		nndivdvds 11816	. . . . 5 |- ( ( B e. NN /\ ( A gcd B ) e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. NN ) )
38	22, 9, 37	syl2anc 411	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) || B <-> ( B / ( A gcd B ) ) e. NN ) )
39	36, 38	mpbid 147	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( B / ( A gcd B ) ) e. NN )
40		qnumdenbi 12205	. . 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 1248	. 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 944	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 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   =/= wne 2357   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   CCcc 7822   0cc0 7824   1c1 7825   # cap 8551   / cdiv 8642   NNcn 8932   ZZcz 9266   QQcq 9632   || cdvds 11807   gcd cgcd 11956  numercnumer 12194  denomcdenom 12195

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-frec 6405  df-sup 6996  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-fz 10022  df-fzo 10156  df-fl 10283  df-mod 10336  df-seqfrec 10459  df-exp 10533  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-dvds 11808  df-gcd 11957  df-numer 12196  df-denom 12197

This theorem is referenced by:  divdenle  12210