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Theorem nndivtr 8730
Description: Transitive property of divisibility: if  A divides  B and  B divides  C, then  A divides  C. Typically,  C would be an integer, although the theorem holds for complex  C. (Contributed by NM, 3-May-2005.)
Assertion
Ref Expression
nndivtr  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  /\  ( ( B  /  A )  e.  NN  /\  ( C  /  B
)  e.  NN ) )  ->  ( C  /  A )  e.  NN )

Proof of Theorem nndivtr
StepHypRef Expression
1 nnmulcl 8709 . . 3  |-  ( ( ( B  /  A
)  e.  NN  /\  ( C  /  B
)  e.  NN )  ->  ( ( B  /  A )  x.  ( C  /  B
) )  e.  NN )
2 nncn 8696 . . . . . . 7  |-  ( B  e.  NN  ->  B  e.  CC )
323ad2ant2 988 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  B  e.  CC )
4 simp3 968 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  C  e.  CC )
5 nncn 8696 . . . . . . . 8  |-  ( A  e.  NN  ->  A  e.  CC )
6 nnap0 8717 . . . . . . . 8  |-  ( A  e.  NN  ->  A #  0 )
75, 6jca 304 . . . . . . 7  |-  ( A  e.  NN  ->  ( A  e.  CC  /\  A #  0 ) )
873ad2ant1 987 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  ( A  e.  CC  /\  A #  0 ) )
9 nnap0 8717 . . . . . . . 8  |-  ( B  e.  NN  ->  B #  0 )
102, 9jca 304 . . . . . . 7  |-  ( B  e.  NN  ->  ( B  e.  CC  /\  B #  0 ) )
11103ad2ant2 988 . . . . . 6  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  ( B  e.  CC  /\  B #  0 ) )
12 divmul24ap 8444 . . . . . 6  |-  ( ( ( B  e.  CC  /\  C  e.  CC )  /\  ( ( A  e.  CC  /\  A #  0 )  /\  ( B  e.  CC  /\  B #  0 ) ) )  ->  ( ( B  /  A )  x.  ( C  /  B
) )  =  ( ( B  /  B
)  x.  ( C  /  A ) ) )
133, 4, 8, 11, 12syl22anc 1202 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
( B  /  A
)  x.  ( C  /  B ) )  =  ( ( B  /  B )  x.  ( C  /  A
) ) )
142, 9dividapd 8514 . . . . . . 7  |-  ( B  e.  NN  ->  ( B  /  B )  =  1 )
1514oveq1d 5757 . . . . . 6  |-  ( B  e.  NN  ->  (
( B  /  B
)  x.  ( C  /  A ) )  =  ( 1  x.  ( C  /  A
) ) )
16153ad2ant2 988 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
( B  /  B
)  x.  ( C  /  A ) )  =  ( 1  x.  ( C  /  A
) ) )
17 divclap 8406 . . . . . . . . . 10  |-  ( ( C  e.  CC  /\  A  e.  CC  /\  A #  0 )  ->  ( C  /  A )  e.  CC )
18173expb 1167 . . . . . . . . 9  |-  ( ( C  e.  CC  /\  ( A  e.  CC  /\  A #  0 ) )  ->  ( C  /  A )  e.  CC )
197, 18sylan2 284 . . . . . . . 8  |-  ( ( C  e.  CC  /\  A  e.  NN )  ->  ( C  /  A
)  e.  CC )
2019ancoms 266 . . . . . . 7  |-  ( ( A  e.  NN  /\  C  e.  CC )  ->  ( C  /  A
)  e.  CC )
2120mulid2d 7752 . . . . . 6  |-  ( ( A  e.  NN  /\  C  e.  CC )  ->  ( 1  x.  ( C  /  A ) )  =  ( C  /  A ) )
22213adant2 985 . . . . 5  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
1  x.  ( C  /  A ) )  =  ( C  /  A ) )
2313, 16, 223eqtrd 2154 . . . 4  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
( B  /  A
)  x.  ( C  /  B ) )  =  ( C  /  A ) )
2423eleq1d 2186 . . 3  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
( ( B  /  A )  x.  ( C  /  B ) )  e.  NN  <->  ( C  /  A )  e.  NN ) )
251, 24syl5ib 153 . 2  |-  ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  ->  (
( ( B  /  A )  e.  NN  /\  ( C  /  B
)  e.  NN )  ->  ( C  /  A )  e.  NN ) )
2625imp 123 1  |-  ( ( ( A  e.  NN  /\  B  e.  NN  /\  C  e.  CC )  /\  ( ( B  /  A )  e.  NN  /\  ( C  /  B
)  e.  NN ) )  ->  ( C  /  A )  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   CCcc 7586   0cc0 7588   1c1 7589    x. cmul 7593   # cap 8311    / cdiv 8400   NNcn 8688
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-po 4188  df-iso 4189  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8305  df-ap 8312  df-div 8401  df-inn 8689
This theorem is referenced by:  permnn  10485
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