Theorem axmulgt0 7970

Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7870 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)

Assertion

Ref	Expression
axmulgt0	|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Proof of Theorem axmulgt0

Step	Hyp	Ref	Expression
1		ax-pre-mulgt0 7870	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
2		0re 7899	. . . 4 |- 0 e. RR
3		ltxrlt 7964	. . . 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		ltxrlt 7964	. . . 4 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> 0 <RR B ) )
6	2, 5	mpan 421	. . 3 |- ( B e. RR -> ( 0 < B <-> 0 <RR B ) )
7	4, 6	bi2anan9 596	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
8		remulcl 7881	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
9		ltxrlt 7964	. . 3 |- ( ( 0 e. RR /\ ( A x. B ) e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
10	2, 8, 9	sylancr 411	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
11	1, 7, 10	3imtr4d 202	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   <RR cltrr 7757   x. cmul 7758   < clt 7933

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850  ax-mulrcl 7852  ax-rnegex 7862  ax-pre-mulgt0 7870

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-ltxr 7938

This theorem is referenced by:  mulgt0  7973  mulgt0i  8008  sin02gt0  11704  sinq12gt0  13401