Theorem nndivtr 9152

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 9131	. . 3 |- ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( ( B / A ) x. ( C / B ) ) e. NN )
2		nncn 9118	. . . . . . 7 |- ( B e. NN -> B e. CC )
3	2	3ad2ant2 1043	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> B e. CC )
4		simp3 1023	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> C e. CC )
5		nncn 9118	. . . . . . . 8 |- ( A e. NN -> A e. CC )
6		nnap0 9139	. . . . . . . 8 |- ( A e. NN -> A # 0 )
7	5, 6	jca 306	. . . . . . 7 |- ( A e. NN -> ( A e. CC /\ A # 0 ) )
8	7	3ad2ant1 1042	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( A e. CC /\ A # 0 ) )
9		nnap0 9139	. . . . . . . 8 |- ( B e. NN -> B # 0 )
10	2, 9	jca 306	. . . . . . 7 |- ( B e. NN -> ( B e. CC /\ B # 0 ) )
11	10	3ad2ant2 1043	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( B e. CC /\ B # 0 ) )
12		divmul24ap 8863	. . . . . 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 1272	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( ( B / B ) x. ( C / A ) ) )
14	2, 9	dividapd 8933	. . . . . . 7 |- ( B e. NN -> ( B / B ) = 1 )
15	14	oveq1d 6016	. . . . . 6 |- ( B e. NN -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
16	15	3ad2ant2 1043	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
17		divclap 8825	. . . . . . . . . 10 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
18	17	3expb 1228	. . . . . . . . 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 8165	. . . . . 6 |- ( ( A e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
22	21	3adant2 1040	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
23	13, 16, 22	3eqtrd 2266	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( C / A ) )
24	23	eleq1d 2298	. . 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 1002   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   CCcc 7997   0cc0 7999   1c1 8000   x. cmul 8004   # cap 8728   / cdiv 8819   NNcn 9110

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111

This theorem is referenced by:  permnn  10993  infpnlem1  12882