Theorem divnumden 12215

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 9291	. . . . 5 |- ( B e. NN -> B e. ZZ )
3	2	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
4		nnne0 8966	. . . . . . . 8 |- ( B e. NN -> B =/= 0 )
5	4	neneqd 2381	. . . . . . 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 11982	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
9	1, 3, 7, 8	syl21anc 1248	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. NN )
10		gcddvds 11983	. . . . 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 12039	. . . 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 1252	. . 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 8952	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. CC )
15	9	nnap0d 8984	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) # 0 )
16	14, 15	dividapd 8762	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
17	13, 16	eqtr3d 2224	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
18		zcn 9277	. . . 4 |- ( A e. ZZ -> A e. CC )
19	18	adantr 276	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
20		nncn 8946	. . . 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 8984	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B # 0 )
24		divcanap7 8697	. . . 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 2195	. . 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 1258	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
27		znq 9643	. . 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 11986	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
30	29	nn0zd 9392	. . . . . 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 8983	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) =/= 0 )
33		dvdsval2 11816	. . . . 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 1249	. . . 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 11822	. . . . 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 12211	. . 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 1249	. 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 945	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 980   = wceq 1364   e. wcel 2160   =/= wne 2360   class class class wbr 4018   ` cfv 5231  (class class class)co 5891   CCcc 7828   0cc0 7830   1c1 7831   # cap 8557   / cdiv 8648   NNcn 8938   ZZcz 9272   QQcq 9638   || cdvds 11813   gcd cgcd 11962  numercnumer 12200  denomcdenom 12201

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-frec 6410  df-sup 7002  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-fz 10028  df-fzo 10162  df-fl 10289  df-mod 10342  df-seqfrec 10465  df-exp 10539  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-dvds 11814  df-gcd 11963  df-numer 12202  df-denom 12203

This theorem is referenced by:  divdenle  12216