Theorem recgt0 9170

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 8443	. . . . 5 |- 0 < 1
2		0re 8316	. . . . . 6 |- 0 e. RR
3		1re 8315	. . . . . 6 |- 1 e. RR
4	2, 3	ltnsymi 8415	. . . . 5 |- ( 0 < 1 -> -. 1 < 0 )
5	1, 4	ax-mp 5	. . . 4 |- -. 1 < 0
6		simpll 531	. . . . . . . . . 10 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> A e. RR )
7		gt0ap0 8944	. . . . . . . . . . 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 9154	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. RR )
10	9	renegcld 8697	. . . . . . . 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 9154	. . . . . . . . . . 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 8833	. . . . . . . . 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 533	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < A )
18	10, 6, 16, 17	mulgt0d 8439	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < ( -u ( 1 / A ) x. A ) )
19	12	recnd 8344	. . . . . . . . . . 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 8999	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) e. CC )
22	20, 8, 21	syl2anc 415	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. CC )
23	22, 20	mulneg1d 8728	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u ( ( 1 / A ) x. A ) )
24		recidap2 9007	. . . . . . . . . 10 |- ( ( A e. CC /\ A # 0 ) -> ( ( 1 / A ) x. A ) = 1 )
25	20, 8, 24	syl2anc 415	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( ( 1 / A ) x. A ) = 1 )
26	25	negeqd 8511	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> -u ( ( 1 / A ) x. A ) = -u 1 )
27	23, 26	eqtrd 2271	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u 1 )
28	18, 27	breqtrd 4151	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < -u 1 )
29		1red 8331	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 1 e. RR )
30	29	lt0neg1d 8833	. . . . . 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 674	. . 3 |- ( ( A e. RR /\ 0 < A ) -> -. ( 1 / A ) < 0 )
34		lenlt 8391	. . . 4 |- ( ( 0 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 <_ ( 1 / A ) <-> -. ( 1 / A ) < 0 ) )
35	2, 13, 34	sylancr 418	. . 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 9005	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( 1 / A ) # 0 )
38	19, 7, 37	syl2anc 415	. . 3 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) # 0 )
39	19, 7, 21	syl2anc 415	. . . 4 |- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. CC )
40		0cn 8308	. . . 4 |- 0 e. CC
41		apsym 8924	. . . 4 |- ( ( ( 1 / A ) e. CC /\ 0 e. CC ) -> ( ( 1 / A ) # 0 <-> 0 # ( 1 / A ) ) )
42	39, 40, 41	sylancl 417	. . 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 8950	. . 3 |- ( ( 0 e. RR /\ ( 1 / A ) e. RR ) -> ( 0 < ( 1 / A ) <-> ( 0 <_ ( 1 / A ) /\ 0 # ( 1 / A ) ) ) )
45	2, 13, 44	sylancr 418	. 2 |- ( ( A e. RR /\ 0 < A ) -> ( 0 < ( 1 / A ) <-> ( 0 <_ ( 1 / A ) /\ 0 # ( 1 / A ) ) ) )
46	36, 43, 45	mpbir2and 957	1 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   x. cmul 8174   < clt 8350   <_ cle 8351   -ucneg 8488   # cap 8899   / cdiv 8992

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993

This theorem is referenced by:  prodgt0gt0  9171  ltdiv1  9188  ltrec1  9208  lerec2  9209  lediv12a  9214  recgt1i  9218  recreclt  9220  recgt0i  9226  recgt0ii  9227  recgt0d  9254  nnrecgt0  9321  nnrecl  9540