Theorem gtndiv 9348

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 8926	. . . 4 |- ( B e. NN -> B e. RR )
2	1	3ad2ant2 1019	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR )
3		simp1 997	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR )
4		nngt0 8944	. . . 4 |- ( B e. NN -> 0 < B )
5	4	3ad2ant2 1019	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < B )
6	4	adantl 277	. . . . 5 |- ( ( A e. RR /\ B e. NN ) -> 0 < B )
7		0re 7957	. . . . . . . 8 |- 0 e. RR
8		lttr 8031	. . . . . . . 8 |- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
9	7, 8	mp3an1 1324	. . . . . . 7 |- ( ( B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
10	1, 9	sylan 283	. . . . . 6 |- ( ( B e. NN /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
11	10	ancoms 268	. . . . 5 |- ( ( A e. RR /\ B e. NN ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
12	6, 11	mpand 429	. . . 4 |- ( ( A e. RR /\ B e. NN ) -> ( B < A -> 0 < A ) )
13	12	3impia 1200	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A )
14	2, 3, 5, 13	divgt0d 8892	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) )
15		simp3 999	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A )
16		1re 7956	. . . . . . 7 |- 1 e. RR
17		ltdivmul2 8835	. . . . . . 7 |- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
18	16, 17	mp3an2 1325	. . . . . 6 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
19	2, 3, 13, 18	syl12anc 1236	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
20		recn 7944	. . . . . . . 8 |- ( A e. RR -> A e. CC )
21	20	mulid2d 7976	. . . . . . 7 |- ( A e. RR -> ( 1 x. A ) = A )
22	21	breq2d 4016	. . . . . 6 |- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) )
23	22	3ad2ant1 1018	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B < ( 1 x. A ) <-> B < A ) )
24	19, 23	bitrd 188	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < A ) )
25	15, 24	mpbird 167	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < 1 )
26		0p1e1 9033	. . 3 |- ( 0 + 1 ) = 1
27	25, 26	breqtrrdi 4046	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) )
28		0z 9264	. . 3 |- 0 e. ZZ
29		btwnnz 9347	. . 3 |- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
30	28, 29	mp3an1 1324	. 2 |- ( ( 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
31	14, 27, 30	syl2anc 411	1 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> -. ( B / A ) e. ZZ )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4004  (class class class)co 5875   RRcr 7810   0cc0 7811   1c1 7812   + caddc 7814   x. cmul 7816   < clt 7992   / cdiv 8629   NNcn 8919   ZZcz 9253

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-mulrcl 7910  ax-addcom 7911  ax-mulcom 7912  ax-addass 7913  ax-mulass 7914  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-1rid 7918  ax-0id 7919  ax-rnegex 7920  ax-precex 7921  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-apti 7926  ax-pre-ltadd 7927  ax-pre-mulgt0 7928  ax-pre-mulext 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-id 4294  df-po 4297  df-iso 4298  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-reap 8532  df-ap 8539  df-div 8630  df-inn 8920  df-n0 9177  df-z 9254

This theorem is referenced by:  prime  9352