Theorem recgt0 9008

Description: The reciprocal of a positive number is positive. Exercise 4 of [Apostol] p. 21. (Contributed by NM, 25-Aug-1999.) (Revised by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
recgt0	|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )

Proof of Theorem recgt0

Step	Hyp	Ref	Expression
1		0lt1 8284	. . . . 5 |- 0 < 1
2		0re 8157	. . . . . 6 |- 0 e. RR
3		1re 8156	. . . . . 6 |- 1 e. RR
4	2, 3	ltnsymi 8257	. . . . 5 |- ( 0 < 1 -> -. 1 < 0 )
5	1, 4	ax-mp 5	. . . 4 |- -. 1 < 0
6		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> A e. RR )
7		gt0ap0 8784	. . . . . . . . . . 11 |- ( ( A e. RR /\ 0 < A ) -> A # 0 )
8	7	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> A # 0 )
9	6, 8	rerecclapd 8992	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. RR )
10	9	renegcld 8537	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> -u ( 1 / A ) e. RR )
11		simpr 110	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) < 0 )
12		simpl 109	. . . . . . . . . . . 12 |- ( ( A e. RR /\ 0 < A ) -> A e. RR )
13	12, 7	rerecclapd 8992	. . . . . . . . . . 11 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR )
14	13	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. RR )
15	14	lt0neg1d 8673	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( ( 1 / A ) < 0 <-> 0 < -u ( 1 / A ) ) )
16	11, 15	mpbid 147	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < -u ( 1 / A ) )
17		simplr 528	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < A )
18	10, 6, 16, 17	mulgt0d 8280	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < ( -u ( 1 / A ) x. A ) )
19	12	recnd 8186	. . . . . . . . . . 11 |- ( ( A e. RR /\ 0 < A ) -> A e. CC )
20	19	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> A e. CC )
21		recclap 8837	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) e. CC )
22	20, 8, 21	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. CC )
23	22, 20	mulneg1d 8568	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u ( ( 1 / A ) x. A ) )
24		recidap2 8845	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 ) -> ( ( 1 / A ) x. A ) = 1 )
25	20, 8, 24	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( ( 1 / A ) x. A ) = 1 )
26	25	negeqd 8352	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> -u ( ( 1 / A ) x. A ) = -u 1 )
27	23, 26	eqtrd 2262	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u 1 )
28	18, 27	breqtrd 4109	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < -u 1 )
29		1red 8172	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 1 e. RR )
30	29	lt0neg1d 8673	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 < 0 <-> 0 < -u 1 ) )
31	28, 30	mpbird 167	. . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 1 < 0 )
32	31	ex 115	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) < 0 -> 1 < 0 ) )
33	5, 32	mtoi 668	. . 3 |- ( ( A e. RR /\ 0 < A ) -> -. ( 1 / A ) < 0 )
34		lenlt 8233	. . . 4 |- ( ( 0 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 <_ ( 1 / A ) <-> -. ( 1 / A ) < 0 ) )
35	2, 13, 34	sylancr 414	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 0 <_ ( 1 / A ) <-> -. ( 1 / A ) < 0 ) )
36	33, 35	mpbird 167	. 2 |- ( ( A e. RR /\ 0 < A ) -> 0 <_ ( 1 / A ) )
37		recap0 8843	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) # 0 )
38	19, 7, 37	syl2anc 411	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) # 0 )
39	19, 7, 21	syl2anc 411	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
40		0cn 8149	. . . 4 |- 0 e. CC
41		apsym 8764	. . . 4 |- ( ( ( 1 / A ) e. CC /\ 0 e. CC ) -> ( ( 1 / A ) # 0 <-> 0 # ( 1 / A ) ) )
42	39, 40, 41	sylancl 413	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( ( 1 / A ) # 0 <-> 0 # ( 1 / A ) ) )
43	38, 42	mpbid 147	. 2 |- ( ( A e. RR /\ 0 < A ) -> 0 # ( 1 / A ) )
44		ltleap 8790	. . 3 |- ( ( 0 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < ( 1 / A ) <-> ( 0 <_ ( 1 / A ) /\ 0 # ( 1 / A ) ) ) )
45	2, 13, 44	sylancr 414	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> ( 0 <_ ( 1 / A ) /\ 0 # ( 1 / A ) ) ) )
46	36, 43, 45	mpbir2and 950	1 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   class class class wbr 4083  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   1c1 8011   x. cmul 8015   < clt 8192   <_ cle 8193   -ucneg 8329   # cap 8739   / cdiv 8830

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831

This theorem is referenced by:  prodgt0gt0  9009  ltdiv1  9026  ltrec1  9046  lerec2  9047  lediv12a  9052  recgt1i  9056  recreclt  9058  recgt0i  9064  recgt0ii  9065  recgt0d  9092  nnrecgt0  9159  nnrecl  9378