Theorem recexgt0 8499

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 7884	. 2 |- ( ( A e. RR /\ 0 <RR A ) -> E. x e. RR ( 0 <RR x /\ ( A x. x ) = 1 ) )
2		0re 7920	. . . 4 |- 0 e. RR
3		ltxrlt 7985	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 422	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5	4	pm5.32i 451	. 2 |- ( ( A e. RR /\ 0 < A ) <-> ( A e. RR /\ 0 <RR A ) )
6		ltxrlt 7985	. . . . 5 |- ( ( 0 e. RR /\ x e. RR ) -> ( 0 < x <-> 0 <RR x ) )
7	2, 6	mpan 422	. . . 4 |- ( x e. RR -> ( 0 < x <-> 0 <RR x ) )
8	7	anbi1d 462	. . 3 |- ( x e. RR -> ( ( 0 < x /\ ( A x. x ) = 1 ) <-> ( 0 <RR x /\ ( A x. x ) = 1 ) ) )
9	8	rexbiia 2485	. 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 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775   <RR cltrr 7778   x. cmul 7779   < clt 7954

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883  ax-precex 7884

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-ltxr 7959

This theorem is referenced by:  ltmul1  8511