Theorem nndivtr 8899

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 8878	. . 3 |- ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( ( B / A ) x. ( C / B ) ) e. NN )
2		nncn 8865	. . . . . . 7 |- ( B e. NN -> B e. CC )
3	2	3ad2ant2 1009	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> B e. CC )
4		simp3 989	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> C e. CC )
5		nncn 8865	. . . . . . . 8 |- ( A e. NN -> A e. CC )
6		nnap0 8886	. . . . . . . 8 |- ( A e. NN -> A # 0 )
7	5, 6	jca 304	. . . . . . 7 |- ( A e. NN -> ( A e. CC /\ A # 0 ) )
8	7	3ad2ant1 1008	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( A e. CC /\ A # 0 ) )
9		nnap0 8886	. . . . . . . 8 |- ( B e. NN -> B # 0 )
10	2, 9	jca 304	. . . . . . 7 |- ( B e. NN -> ( B e. CC /\ B # 0 ) )
11	10	3ad2ant2 1009	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( B e. CC /\ B # 0 ) )
12		divmul24ap 8612	. . . . . 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 1229	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( ( B / B ) x. ( C / A ) ) )
14	2, 9	dividapd 8682	. . . . . . 7 |- ( B e. NN -> ( B / B ) = 1 )
15	14	oveq1d 5857	. . . . . 6 |- ( B e. NN -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
16	15	3ad2ant2 1009	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
17		divclap 8574	. . . . . . . . . 10 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
18	17	3expb 1194	. . . . . . . . 9 |- ( ( C e. CC /\ ( A e. CC /\ A # 0 ) ) -> ( C / A ) e. CC )
19	7, 18	sylan2 284	. . . . . . . 8 |- ( ( C e. CC /\ A e. NN ) -> ( C / A ) e. CC )
20	19	ancoms 266	. . . . . . 7 |- ( ( A e. NN /\ C e. CC ) -> ( C / A ) e. CC )
21	20	mulid2d 7917	. . . . . 6 |- ( ( A e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
22	21	3adant2 1006	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
23	13, 16, 22	3eqtrd 2202	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( C / A ) )
24	23	eleq1d 2235	. . 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 153	. 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 123	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 103   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   1c1 7754   x. cmul 7758   # cap 8479   / cdiv 8568   NNcn 8857

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858

This theorem is referenced by:  permnn  10684  infpnlem1  12289