Theorem recexgt0 8853

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 8236	. 2 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2		0re 8273	. . . 4 |- 0 e. RR
3		ltxrlt 8338	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 424	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5	4	pm5.32i 454	. 2 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
6		ltxrlt 8338	. . . . 5 |- ( ( 0 e. RR /\ x e. RR ) -> ( 0 < x <-> 0 <RR x ) )
7	2, 6	mpan 424	. . . 4 |- ( x e. RR -> ( 0 < x <-> 0 <RR x ) )
8	7	anbi1d 465	. . 3 |- ( x e. RR -> ( ( 0 < x /\ ( A x. x ) = 1 ) <-> ( 0 <RR x /\ ( A x. x ) = 1 ) ) )
9	8	rexbiia 2557	. 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 201	1 |- ( ( A e. RR /\ 0 < A ) -> E. x e. RR ( 0 < x /\ ( A x. x ) = 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   E.wrex 2521   class class class wbr 4108  (class class class)co 6049   RRcr 8125   0cc0 8126   1c1 8127   <RR cltrr 8130   x. cmul 8131   < clt 8307

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8217  ax-resscn 8218  ax-1re 8220  ax-addrcl 8223  ax-rnegex 8235  ax-precex 8236

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-xp 4754  df-pnf 8309  df-mnf 8310  df-ltxr 8312

This theorem is referenced by:  ltmul1  8865