Theorem divnumden 12901

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 9601	. . . . 5 |- ( B e. NN -> B e. ZZ )
3	2	adantl 277	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> B e. ZZ )
4		nnne0 9270	. . . . . . . 8 |- ( B e. NN -> B =/= 0 )
5	4	neneqd 2435	. . . . . . 7 |- ( B e. NN -> -. B = 0 )
6	5	adantl 277	. . . . . 6 |- ( ( A e. ZZ /\ B e. NN ) -> -. B = 0 )
7	6	intnand 939	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> -. ( A = 0 /\ B = 0 ) )
8		gcdn0cl 12666	. . . . 5 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 /\ B = 0 ) ) -> ( A gcd B ) e. NN )
9	1, 3, 7, 8	syl21anc 1273	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. NN )
10		gcddvds 12667	. . . . 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 12723	. . . 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 1277	. . 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 9256	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) e. CC )
15	9	nnap0d 9288	. . . 4 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) # 0 )
16	14, 15	dividapd 9065	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A gcd B ) / ( A gcd B ) ) = 1 )
17	13, 16	eqtr3d 2269	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( ( A / ( A gcd B ) ) gcd ( B / ( A gcd B ) ) ) = 1 )
18		zcn 9587	. . . 4 |- ( A e. ZZ -> A e. CC )
19	18	adantr 276	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> A e. CC )
20		nncn 9250	. . . 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 9288	. . 3 |- ( ( A e. ZZ /\ B e. NN ) -> B # 0 )
24		divcanap7 9000	. . . 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 2240	. . 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 1283	. 2 |- ( ( A e. ZZ /\ B e. NN ) -> ( A / B ) = ( ( A / ( A gcd B ) ) / ( B / ( A gcd B ) ) ) )
27		znq 9962	. . 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 12670	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) e. NN0 )
30	29	nn0zd 9704	. . . . . 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 9287	. . . . 5 |- ( ( A e. ZZ /\ B e. NN ) -> ( A gcd B ) =/= 0 )
33		dvdsval2 12484	. . . . 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 1274	. . . 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 12490	. . . . 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 12897	. . 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 1274	. 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 952	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 1005   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   CCcc 8130   0cc0 8132   1c1 8133   # cap 8860   / cdiv 8951   NNcn 9242   ZZcz 9582   QQcq 9957   || cdvds 12481   gcd cgcd 12657  numercnumer 12886  denomcdenom 12887

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250  ax-arch 8251  ax-caucvg 8252

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-frec 6624  df-sup 7277  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-2 9301  df-3 9302  df-4 9303  df-n0 9502  df-z 9583  df-uz 9860  df-q 9958  df-rp 9993  df-fz 10349  df-fzo 10484  df-fl 10637  df-mod 10692  df-seqfrec 10817  df-exp 10908  df-cj 11535  df-re 11536  df-im 11537  df-rsqrt 11691  df-abs 11692  df-dvds 12482  df-gcd 12658  df-numer 12888  df-denom 12889

This theorem is referenced by:  divdenle  12902