Theorem divnumden 12761

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 9491	. . . . 5 |- ( B e. NN -> B e. ZZ )
3	2	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
4		nnne0 9164	. . . . . . . 8 |- ( B e. NN -> B =/= 0 )
5	4	neneqd 2421	. . . . . . 7 |- ( B e. NN -> -. B = 0 )
6	5	adantl 277	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> -. B = 0 )
7	6	intnand 936	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
8		gcdn0cl 12526	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
9	1, 3, 7, 8	syl21anc 1270	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. NN )
10		gcddvds 12527	. . . . 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 12583	. . . 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 1274	. . 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 9150	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. CC )
15	9	nnap0d 9182	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) # 0 )
16	14, 15	dividapd 8959	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
17	13, 16	eqtr3d 2264	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
18		zcn 9477	. . . 4 |- ( A e. ZZ -> A e. CC )
19	18	adantr 276	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
20		nncn 9144	. . . 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 9182	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B # 0 )
24		divcanap7 8894	. . . 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 2235	. . 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 1280	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
27		znq 9851	. . 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 12530	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
30	29	nn0zd 9593	. . . . . 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 9181	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) =/= 0 )
33		dvdsval2 12344	. . . . 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 1271	. . . 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 12350	. . . . 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 12757	. . 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 1271	. 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 949	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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8023   0cc0 8025   1c1 8026   # cap 8754   / cdiv 8845   NNcn 9136   ZZcz 9472   QQcq 9846   || cdvds 12341   gcd cgcd 12517  numercnumer 12746  denomcdenom 12747

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7177  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-gcd 12518  df-numer 12748  df-denom 12749

This theorem is referenced by:  divdenle  12762