Theorem gtndiv 9286

Description: A larger number does not divide a smaller positive integer. (Contributed by NM, 3-May-2005.)

Assertion

Ref	Expression
gtndiv	|- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ )

Proof of Theorem gtndiv

Step	Hyp	Ref	Expression
1		nnre 8864	. . . 4 |- ( B e. NN -> B e. RR )
2	1	3ad2ant2 1009	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR )
3		simp1 987	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR )
4		nngt0 8882	. . . 4 |- ( B e. NN -> 0 < B )
5	4	3ad2ant2 1009	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < B )
6	4	adantl 275	. . . . 5 |- ( ( A e. RR /\ B e. NN ) -> 0 < B )
7		0re 7899	. . . . . . . 8 |- 0 e. RR
8		lttr 7972	. . . . . . . 8 |- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
9	7, 8	mp3an1 1314	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
10	1, 9	sylan 281	. . . . . 6 |- ( ( B e. NN /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
11	10	ancoms 266	. . . . 5 |- ( ( A e. RR /\ B e. NN ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
12	6, 11	mpand 426	. . . 4 |- ( ( A e. RR /\ B e. NN ) -> ( B < A -> 0 < A ) )
13	12	3impia 1190	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A )
14	2, 3, 5, 13	divgt0d 8830	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) )
15		simp3 989	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A )
16		1re 7898	. . . . . . 7 |- 1 e. RR
17		ltdivmul2 8773	. . . . . . 7 |- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
18	16, 17	mp3an2 1315	. . . . . 6 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
19	2, 3, 13, 18	syl12anc 1226	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
20		recn 7886	. . . . . . . 8 |- ( A e. RR -> A e. CC )
21	20	mulid2d 7917	. . . . . . 7 |- ( A e. RR -> ( 1 x. A ) = A )
22	21	breq2d 3994	. . . . . 6 |- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) )
23	22	3ad2ant1 1008	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B < ( 1 x. A ) <-> B < A ) )
24	19, 23	bitrd 187	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < A ) )
25	15, 24	mpbird 166	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < 1 )
26		0p1e1 8971	. . 3 |- ( 0 + 1 ) = 1
27	25, 26	breqtrrdi 4024	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) )
28		0z 9202	. . 3 |- 0 e. ZZ
29		btwnnz 9285	. . 3 |- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
30	28, 29	mp3an1 1314	. 2 |- ( ( 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
31	14, 27, 30	syl2anc 409	1 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   / cdiv 8568   NNcn 8857   ZZcz 9191

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-3or 969  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  df-n0 9115  df-z 9192

This theorem is referenced by:  prime  9290