Theorem mulgt0ii 8430

Description: The product of two positive numbers is positive. (Contributed by NM, 18-May-1999.)

Hypotheses

Ref	Expression
lt.1	|- A e. RR
lt.2	|- B e. RR
mulgt0i.3	|- 0 < A
mulgt0i.4	|- 0 < B

Assertion

Ref	Expression
mulgt0ii	|- 0 < ( A x. B )

Proof of Theorem mulgt0ii

Step	Hyp	Ref	Expression
1		mulgt0i.3	. 2 |- 0 < A
2		mulgt0i.4	. 2 |- 0 < B
3		lt.1	. . 3 |- A e. RR
4		lt.2	. . 3 |- B e. RR
5	3, 4	mulgt0i 8429	. 2 |- ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) )
6	1, 2, 5	mp2an 430	1 |- 0 < ( A x. B )

Syntax hints:   e. wcel 2209   class class class wbr 4128  (class class class)co 6079   RRcr 8172   0cc0 8173   x. cmul 8178   < clt 8354

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 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270  ax-mulrcl 8272  ax-rnegex 8282  ax-pre-mulgt0 8290

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 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-xp 4778  df-pnf 8356  df-mnf 8357  df-ltxr 8359

This theorem is referenced by:  ef01bndlem  12506