Theorem gtndiv 9438

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 9014	. . . 4 |- ( B e. NN -> B e. RR )
2	1	3ad2ant2 1021	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR )
3		simp1 999	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR )
4		nngt0 9032	. . . 4 |- ( B e. NN -> 0 < B )
5	4	3ad2ant2 1021	. . 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 8043	. . . . . . . 8 |- 0 e. RR
8		lttr 8117	. . . . . . . 8 |- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
9	7, 8	mp3an1 1335	. . . . . . 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 1202	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A )
14	2, 3, 5, 13	divgt0d 8979	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) )
15		simp3 1001	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A )
16		1re 8042	. . . . . . 7 |- 1 e. RR
17		ltdivmul2 8922	. . . . . . 7 |- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
18	16, 17	mp3an2 1336	. . . . . 6 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
19	2, 3, 13, 18	syl12anc 1247	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
20		recn 8029	. . . . . . . 8 |- ( A e. RR -> A e. CC )
21	20	mulid2d 8062	. . . . . . 7 |- ( A e. RR -> ( 1 x. A ) = A )
22	21	breq2d 4046	. . . . . 6 |- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) )
23	22	3ad2ant1 1020	. . . . 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 9121	. . 3 |- ( 0 + 1 ) = 1
27	25, 26	breqtrrdi 4076	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) )
28		0z 9354	. . 3 |- 0 e. ZZ
29		btwnnz 9437	. . 3 |- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
30	28, 29	mp3an1 1335	. 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 980   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   RRcr 7895   0cc0 7896   1c1 7897   + caddc 7899   x. cmul 7901   < clt 8078   / cdiv 8716   NNcn 9007   ZZcz 9343

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344

This theorem is referenced by:  prime  9442