Theorem recgt0 9029

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 8305	. . . . 5 |- 0 < 1
2		0re 8178	. . . . . 6 |- 0 e. RR
3		1re 8177	. . . . . 6 |- 1 e. RR
4	2, 3	ltnsymi 8278	. . . . 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 8805	. . . . . . . . . . 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 9013	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. RR )
10	9	renegcld 8558	. . . . . . . 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 9013	. . . . . . . . . . 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 8694	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < A )
18	10, 6, 16, 17	mulgt0d 8301	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < ( -u ( 1 / A ) x. A ) )
19	12	recnd 8207	. . . . . . . . . . 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 8858	. . . . . . . . . 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 8589	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u ( ( 1 / A ) x. A ) )
24		recidap2 8866	. . . . . . . . . 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 8373	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> -u ( ( 1 / A ) x. A ) = -u 1 )
27	23, 26	eqtrd 2264	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u 1 )
28	18, 27	breqtrd 4114	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < -u 1 )
29		1red 8193	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 1 e. RR )
30	29	lt0neg1d 8694	. . . . . 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 670	. . 3 |- ( ( A e. RR /\ 0 < A ) -> -. ( 1 / A ) < 0 )
34		lenlt 8254	. . . 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 8864	. . . 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 8170	. . . 4 |- 0 e. CC
41		apsym 8785	. . . 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 8811	. . 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 952	1 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   CCcc 8029   RRcr 8030   0cc0 8031   1c1 8032   x. cmul 8036   < clt 8213   <_ cle 8214   -ucneg 8350   # cap 8760   / cdiv 8851

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  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-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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852

This theorem is referenced by:  prodgt0gt0  9030  ltdiv1  9047  ltrec1  9067  lerec2  9068  lediv12a  9073  recgt1i  9077  recreclt  9079  recgt0i  9085  recgt0ii  9086  recgt0d  9113  nnrecgt0  9180  nnrecl  9399