Theorem axmulgt0 8387

Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 8286 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 8286	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
2		0re 8316	. . . 4 |- 0 e. RR
3		ltxrlt 8381	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 428	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5		ltxrlt 8381	. . . 4 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> 0 <RR B ) )
6	2, 5	mpan 428	. . 3 |- ( B e. RR -> ( 0 < B <-> 0 <RR B ) )
7	4, 6	bi2anan9 614	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
8		remulcl 8297	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
9		ltxrlt 8381	. . 3 |- ( ( 0 e. RR /\ ( A x. B ) e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
10	2, 8, 9	sylancr 418	. 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 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   <RR cltrr 8173   x. cmul 8174   < clt 8350

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-mulrcl 8268  ax-rnegex 8278  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-pnf 8352  df-mnf 8353  df-ltxr 8355

This theorem is referenced by:  mulgt0  8390  mulgt0i  8425  sin02gt0  12509  sinq12gt0  15854