Theorem gtndiv 9505

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 9080	. . . 4 |- ( B e. NN -> B e. RR )
2	1	3ad2ant2 1022	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR )
3		simp1 1000	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR )
4		nngt0 9098	. . . 4 |- ( B e. NN -> 0 < B )
5	4	3ad2ant2 1022	. . 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 8109	. . . . . . . 8 |- 0 e. RR
8		lttr 8183	. . . . . . . 8 |- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
9	7, 8	mp3an1 1337	. . . . . . 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 1203	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A )
14	2, 3, 5, 13	divgt0d 9045	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) )
15		simp3 1002	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A )
16		1re 8108	. . . . . . 7 |- 1 e. RR
17		ltdivmul2 8988	. . . . . . 7 |- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
18	16, 17	mp3an2 1338	. . . . . 6 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
19	2, 3, 13, 18	syl12anc 1248	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
20		recn 8095	. . . . . . . 8 |- ( A e. RR -> A e. CC )
21	20	mulid2d 8128	. . . . . . 7 |- ( A e. RR -> ( 1 x. A ) = A )
22	21	breq2d 4072	. . . . . 6 |- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) )
23	22	3ad2ant1 1021	. . . . 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 9187	. . 3 |- ( 0 + 1 ) = 1
27	25, 26	breqtrrdi 4102	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) )
28		0z 9420	. . 3 |- 0 e. ZZ
29		btwnnz 9504	. . 3 |- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
30	28, 29	mp3an1 1337	. 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 981   e. wcel 2178   class class class wbr 4060  (class class class)co 5969   RRcr 7961   0cc0 7962   1c1 7963   + caddc 7965   x. cmul 7967   < clt 8144   / cdiv 8782   NNcn 9073   ZZcz 9409

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604  ax-cnex 8053  ax-resscn 8054  ax-1cn 8055  ax-1re 8056  ax-icn 8057  ax-addcl 8058  ax-addrcl 8059  ax-mulcl 8060  ax-mulrcl 8061  ax-addcom 8062  ax-mulcom 8063  ax-addass 8064  ax-mulass 8065  ax-distr 8066  ax-i2m1 8067  ax-0lt1 8068  ax-1rid 8069  ax-0id 8070  ax-rnegex 8071  ax-precex 8072  ax-cnre 8073  ax-pre-ltirr 8074  ax-pre-ltwlin 8075  ax-pre-lttrn 8076  ax-pre-apti 8077  ax-pre-ltadd 8078  ax-pre-mulgt0 8079  ax-pre-mulext 8080

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2779  df-sbc 3007  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-int 3901  df-br 4061  df-opab 4123  df-id 4359  df-po 4362  df-iso 4363  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-iota 5252  df-fun 5293  df-fv 5299  df-riota 5924  df-ov 5972  df-oprab 5973  df-mpo 5974  df-pnf 8146  df-mnf 8147  df-xr 8148  df-ltxr 8149  df-le 8150  df-sub 8282  df-neg 8283  df-reap 8685  df-ap 8692  df-div 8783  df-inn 9074  df-n0 9333  df-z 9410

This theorem is referenced by:  prime  9509