Theorem axmulgt0 8029

Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7928 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 7928	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
2		0re 7957	. . . 4 |- 0 e. RR
3		ltxrlt 8023	. . . 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		ltxrlt 8023	. . . 4 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> 0 <RR B ) )
6	2, 5	mpan 424	. . 3 |- ( B e. RR -> ( 0 < B <-> 0 <RR B ) )
7	4, 6	bi2anan9 606	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
8		remulcl 7939	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
9		ltxrlt 8023	. . 3 |- ( ( 0 e. RR /\ ( A x. B ) e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
10	2, 8, 9	sylancr 414	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
11	1, 7, 10	3imtr4d 203	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2148   class class class wbr 4004  (class class class)co 5875   RRcr 7810   0cc0 7811   <RR cltrr 7815   x. cmul 7816   < clt 7992

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 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908  ax-mulrcl 7910  ax-rnegex 7920  ax-pre-mulgt0 7928

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 2740  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-xp 4633  df-pnf 7994  df-mnf 7995  df-ltxr 7997

This theorem is referenced by:  mulgt0  8032  mulgt0i  8067  sin02gt0  11771  sinq12gt0  14254