Theorem gtndiv 9575

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 9150	. . . 4 |- ( B e. NN -> B e. RR )
2	1	3ad2ant2 1045	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B e. RR )
3		simp1 1023	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> A e. RR )
4		nngt0 9168	. . . 4 |- ( B e. NN -> 0 < B )
5	4	3ad2ant2 1045	. . 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 8179	. . . . . . . 8 |- 0 e. RR
8		lttr 8253	. . . . . . . 8 |- ( ( 0 e. RR /\ B e. RR /\ A e. RR ) -> ( ( 0 < B /\ B < A ) -> 0 < A ) )
9	7, 8	mp3an1 1360	. . . . . . 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 1226	. . 3 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < A )
14	2, 3, 5, 13	divgt0d 9115	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> 0 < ( B / A ) )
15		simp3 1025	. . . 4 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> B < A )
16		1re 8178	. . . . . . 7 |- 1 e. RR
17		ltdivmul2 9058	. . . . . . 7 |- ( ( B e. RR /\ 1 e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
18	16, 17	mp3an2 1361	. . . . . 6 |- ( ( B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
19	2, 3, 13, 18	syl12anc 1271	. . . . 5 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( ( B / A ) < 1 <-> B < ( 1 x. A ) ) )
20		recn 8165	. . . . . . . 8 |- ( A e. RR -> A e. CC )
21	20	mulid2d 8198	. . . . . . 7 |- ( A e. RR -> ( 1 x. A ) = A )
22	21	breq2d 4100	. . . . . 6 |- ( A e. RR -> ( B < ( 1 x. A ) <-> B < A ) )
23	22	3ad2ant1 1044	. . . . 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 9257	. . 3 |- ( 0 + 1 ) = 1
27	25, 26	breqtrrdi 4130	. 2 |- ( ( A e. RR /\ B e. NN /\ B < A ) -> ( B / A ) < ( 0 + 1 ) )
28		0z 9490	. . 3 |- 0 e. ZZ
29		btwnnz 9574	. . 3 |- ( ( 0 e. ZZ /\ 0 < ( B / A ) /\ ( B / A ) < ( 0 + 1 ) ) -> -. ( B / A ) e. ZZ )
30	28, 29	mp3an1 1360	. 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 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   RRcr 8031   0cc0 8032   1c1 8033   + caddc 8035   x. cmul 8037   < clt 8214   / cdiv 8852   NNcn 9143   ZZcz 9479

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480

This theorem is referenced by:  prime  9579