Theorem recexgt0 8455

Description: Existence of reciprocal of positive real number. (Contributed by Jim Kingdon, 6-Feb-2020.)

Assertion

Ref	Expression
recexgt0	|- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )

Distinct variable group:   x, A

Proof of Theorem recexgt0

Step	Hyp	Ref	Expression
1		ax-precex 7842	. 2 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2		0re 7878	. . . 4 |- 0 e. RR
3		ltxrlt 7943	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 421	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5	4	pm5.32i 450	. 2 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
6		ltxrlt 7943	. . . . 5 |- ( ( 0 e. RR /\ x e. RR ) -> ( 0 < x <-> 0 <RR x ) )
7	2, 6	mpan 421	. . . 4 |- ( x e. RR -> ( 0 < x <-> 0 <RR x ) )
8	7	anbi1d 461	. . 3 |- ( x e. RR -> ( ( 0 < x /\ ( A x. x ) = 1 ) <-> ( 0 <RR x /\ ( A x. x ) = 1 ) ) )
9	8	rexbiia 2472	. 2 |- ( E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) <-> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
10	1, 5, 9	3imtr4i 200	1 |- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733   <RR cltrr 7736   x. cmul 7737   < clt 7912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1re 7826  ax-addrcl 7829  ax-rnegex 7841  ax-precex 7842

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-ltxr 7917

This theorem is referenced by:  ltmul1  8467