Theorem recexgt0 8539

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 7923	. 2 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2		0re 7959	. . . 4 |- 0 e. RR
3		ltxrlt 8025	. . . 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 8025	. . . . 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 2492	. 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 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   1c1 7814   <RR cltrr 7817   x. cmul 7818   < clt 7994

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910  ax-rnegex 7922  ax-precex 7923

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-pnf 7996  df-mnf 7997  df-ltxr 7999

This theorem is referenced by:  ltmul1  8551