Theorem recexgt0 8727

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 8109	. 2 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2		0re 8146	. . . 4 |- 0 e. RR
3		ltxrlt 8212	. . . 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 8212	. . . . 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 2545	. 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 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   <RR cltrr 8003   x. cmul 8004   < clt 8181

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096  ax-rnegex 8108  ax-precex 8109

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8183  df-mnf 8184  df-ltxr 8186

This theorem is referenced by:  ltmul1  8739