Theorem nndivtr 9034

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 9013	. . 3 |- ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( ( B / A ) x. ( C / B ) ) e. NN )
2		nncn 9000	. . . . . . 7 |- ( B e. NN -> B e. CC )
3	2	3ad2ant2 1021	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> B e. CC )
4		simp3 1001	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> C e. CC )
5		nncn 9000	. . . . . . . 8 |- ( A e. NN -> A e. CC )
6		nnap0 9021	. . . . . . . 8 |- ( A e. NN -> A # 0 )
7	5, 6	jca 306	. . . . . . 7 |- ( A e. NN -> ( A e. CC /\ A # 0 ) )
8	7	3ad2ant1 1020	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( A e. CC /\ A # 0 ) )
9		nnap0 9021	. . . . . . . 8 |- ( B e. NN -> B # 0 )
10	2, 9	jca 306	. . . . . . 7 |- ( B e. NN -> ( B e. CC /\ B # 0 ) )
11	10	3ad2ant2 1021	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( B e. CC /\ B # 0 ) )
12		divmul24ap 8745	. . . . . 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 1250	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( ( B / B ) x. ( C / A ) ) )
14	2, 9	dividapd 8815	. . . . . . 7 |- ( B e. NN -> ( B / B ) = 1 )
15	14	oveq1d 5938	. . . . . 6 |- ( B e. NN -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
16	15	3ad2ant2 1021	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
17		divclap 8707	. . . . . . . . . 10 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
18	17	3expb 1206	. . . . . . . . 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 8047	. . . . . 6 |- ( ( A e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
22	21	3adant2 1018	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
23	13, 16, 22	3eqtrd 2233	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( C / A ) )
24	23	eleq1d 2265	. . 3 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( ( B / A ) x. ( C / B ) ) e. NN <-> ( C / A ) e. NN ) )
25	1, 24	imbitrid 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5923   CCcc 7879   0cc0 7881   1c1 7882   x. cmul 7886   # cap 8610   / cdiv 8701   NNcn 8992

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-mulrcl 7980  ax-addcom 7981  ax-mulcom 7982  ax-addass 7983  ax-mulass 7984  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-1rid 7988  ax-0id 7989  ax-rnegex 7990  ax-precex 7991  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997  ax-pre-mulgt0 7998  ax-pre-mulext 7999

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-reap 8604  df-ap 8611  df-div 8702  df-inn 8993

This theorem is referenced by:  permnn  10865  infpnlem1  12538