Theorem nndivtr 8932

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

Step	Hyp	Ref	Expression
1		nnmulcl 8911	. . 3 |- ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( ( B / A ) x. ( C / B ) ) e. NN )
2		nncn 8898	. . . . . . 7 |- ( B e. NN -> B e. CC )
3	2	3ad2ant2 1019	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> B e. CC )
4		simp3 999	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> C e. CC )
5		nncn 8898	. . . . . . . 8 |- ( A e. NN -> A e. CC )
6		nnap0 8919	. . . . . . . 8 |- ( A e. NN -> A # 0 )
7	5, 6	jca 306	. . . . . . 7 |- ( A e. NN -> ( A e. CC /\ A # 0 ) )
8	7	3ad2ant1 1018	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( A e. CC /\ A # 0 ) )
9		nnap0 8919	. . . . . . . 8 |- ( B e. NN -> B # 0 )
10	2, 9	jca 306	. . . . . . 7 |- ( B e. NN -> ( B e. CC /\ B # 0 ) )
11	10	3ad2ant2 1019	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( B e. CC /\ B # 0 ) )
12		divmul24ap 8645	. . . . . 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 ) ) )
13	3, 4, 8, 11, 12	syl22anc 1239	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( ( B / B ) x. ( C / A ) ) )
14	2, 9	dividapd 8715	. . . . . . 7 |- ( B e. NN -> ( B / B ) = 1 )
15	14	oveq1d 5880	. . . . . 6 |- ( B e. NN -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
16	15	3ad2ant2 1019	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
17		divclap 8607	. . . . . . . . . 10 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
18	17	3expb 1204	. . . . . . . . 9 |- ( ( C e. CC /\ ( A e. CC /\ A # 0 ) ) -> ( C / A ) e. CC )
19	7, 18	sylan2 286	. . . . . . . 8 |- ( ( C e. CC /\ A e. NN ) -> ( C / A ) e. CC )
20	19	ancoms 268	. . . . . . 7 |- ( ( A e. NN /\ C e. CC ) -> ( C / A ) e. CC )
21	20	mulid2d 7950	. . . . . 6 |- ( ( A e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
22	21	3adant2 1016	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
23	13, 16, 22	3eqtrd 2212	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( C / A ) )
24	23	eleq1d 2244	. . 3 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( ( B / A ) x. ( C / B ) ) e. NN <-> ( C / A ) e. NN ) )
25	1, 24	syl5ib 154	. 2 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( C / A ) e. NN ) )
26	25	imp 124	1 |- ( ( ( A e. NN /\ B e. NN /\ C e. CC ) /\ ( ( B / A ) e. NN /\ ( C / B ) e. NN ) ) -> ( C / A ) e. NN )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 978   = wceq 1353   e. wcel 2146   class class class wbr 3998  (class class class)co 5865   CCcc 7784   0cc0 7786   1c1 7787   x. cmul 7791   # cap 8512   / cdiv 8601   NNcn 8890

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-mulrcl 7885  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-precex 7896  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-apti 7901  ax-pre-ltadd 7902  ax-pre-mulgt0 7903  ax-pre-mulext 7904

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rmo 2461  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-id 4287  df-po 4290  df-iso 4291  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-reap 8506  df-ap 8513  df-div 8602  df-inn 8891

This theorem is referenced by:  permnn  10717  infpnlem1  12322