Theorem nndivtr 9328

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 9307	. . 3 |- ( ( ( B / A ) e. NN /\ ( C / B ) e. NN ) -> ( ( B / A ) x. ( C / B ) ) e. NN )
2		nncn 9294	. . . . . . 7 |- ( B e. NN -> B e. CC )
3	2	3ad2ant2 1050	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> B e. CC )
4		simp3 1030	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> C e. CC )
5		nncn 9294	. . . . . . . 8 |- ( A e. NN -> A e. CC )
6		nnap0 9315	. . . . . . . 8 |- ( A e. NN -> A # 0 )
7	5, 6	jca 306	. . . . . . 7 |- ( A e. NN -> ( A e. CC /\ A # 0 ) )
8	7	3ad2ant1 1049	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( A e. CC /\ A # 0 ) )
9		nnap0 9315	. . . . . . . 8 |- ( B e. NN -> B # 0 )
10	2, 9	jca 306	. . . . . . 7 |- ( B e. NN -> ( B e. CC /\ B # 0 ) )
11	10	3ad2ant2 1050	. . . . . 6 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( B e. CC /\ B # 0 ) )
12		divmul24ap 9039	. . . . . 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 1279	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( ( B / B ) x. ( C / A ) ) )
14	2, 9	dividapd 9109	. . . . . . 7 |- ( B e. NN -> ( B / B ) = 1 )
15	14	oveq1d 6093	. . . . . 6 |- ( B e. NN -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
16	15	3ad2ant2 1050	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / B ) x. ( C / A ) ) = ( 1 x. ( C / A ) ) )
17		divclap 9001	. . . . . . . . . 10 |- ( ( C e. CC /\ A e. CC /\ A # 0 ) -> ( C / A ) e. CC )
18	17	3expb 1235	. . . . . . . . 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	mullidd 8337	. . . . . 6 |- ( ( A e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
22	21	3adant2 1047	. . . . 5 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( 1 x. ( C / A ) ) = ( C / A ) )
23	13, 16, 22	3eqtrd 2275	. . . 4 |- ( ( A e. NN /\ B e. NN /\ C e. CC ) -> ( ( B / A ) x. ( C / B ) ) = ( C / A ) )
24	23	eleq1d 2307	. . 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4128  (class class class)co 6078   CCcc 8170   0cc0 8172   1c1 8173   x. cmul 8177   # cap 8902   / cdiv 8995   NNcn 9286

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8263  ax-resscn 8264  ax-1cn 8265  ax-1re 8266  ax-icn 8267  ax-addcl 8268  ax-addrcl 8269  ax-mulcl 8270  ax-mulrcl 8271  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-i2m1 8277  ax-0lt1 8278  ax-1rid 8279  ax-0id 8280  ax-rnegex 8281  ax-precex 8282  ax-cnre 8283  ax-pre-ltirr 8284  ax-pre-ltwlin 8285  ax-pre-lttrn 8286  ax-pre-apti 8287  ax-pre-ltadd 8288  ax-pre-mulgt0 8289  ax-pre-mulext 8290

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-po 4439  df-iso 4440  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-iota 5335  df-fun 5377  df-fv 5383  df-riota 6031  df-ov 6081  df-oprab 6082  df-mpo 6083  df-pnf 8355  df-mnf 8356  df-xr 8357  df-ltxr 8358  df-le 8359  df-sub 8492  df-neg 8493  df-reap 8896  df-ap 8903  df-div 8996  df-inn 9287

This theorem is referenced by:  permnn  11191  infpnlem1  13119