Theorem recgt0 8943

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 8219	. . . . 5 |- 0 < 1
2		0re 8092	. . . . . 6 |- 0 e. RR
3		1re 8091	. . . . . 6 |- 1 e. RR
4	2, 3	ltnsymi 8192	. . . . 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 8719	. . . . . . . . . . 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 8927	. . . . . . . . 9 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( 1 / A ) e. RR )
10	9	renegcld 8472	. . . . . . . 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 8927	. . . . . . . . . . 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 8608	. . . . . . . . 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 8215	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < ( -u ( 1 / A ) x. A ) )
19	12	recnd 8121	. . . . . . . . . . 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 8772	. . . . . . . . . 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 8503	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u ( ( 1 / A ) x. A ) )
24		recidap2 8780	. . . . . . . . . 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 8287	. . . . . . . 8 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> -u ( ( 1 / A ) x. A ) = -u 1 )
27	23, 26	eqtrd 2239	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> ( -u ( 1 / A ) x. A ) = -u 1 )
28	18, 27	breqtrd 4077	. . . . . 6 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 0 < -u 1 )
29		1red 8107	. . . . . . 7 |- ( ( ( A e. RR /\ 0 < A ) /\ ( 1 / A ) < 0 ) -> 1 e. RR )
30	29	lt0neg1d 8608	. . . . . 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 666	. . 3 |- ( ( A e. RR /\ 0 < A ) -> -. ( 1 / A ) < 0 )
34		lenlt 8168	. . . 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 8778	. . . 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 8084	. . . 4 |- 0 e. CC
41		apsym 8699	. . . 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 8725	. . 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 947	1 |- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   class class class wbr 4051  (class class class)co 5957   CCcc 7943   RRcr 7944   0cc0 7945   1c1 7946   x. cmul 7950   < clt 8127   <_ cle 8128   -ucneg 8264   # cap 8674   / cdiv 8765

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 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-id 4348  df-po 4351  df-iso 4352  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766

This theorem is referenced by:  prodgt0gt0  8944  ltdiv1  8961  ltrec1  8981  lerec2  8982  lediv12a  8987  recgt1i  8991  recreclt  8993  recgt0i  8999  recgt0ii  9000  recgt0d  9027  nnrecgt0  9094  nnrecl  9313