Theorem mulgt0d 8439

Description: The product of two positive numbers is positive. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
ltd.1	|- ( ph -> A e. RR )
ltd.2	|- ( ph -> B e. RR )
mulgt0d.3	|- ( ph -> 0 < A )
mulgt0d.4	|- ( ph -> 0 < B )

Assertion

Ref	Expression
mulgt0d	|- ( ph -> 0 < ( A x. B ) )

Proof of Theorem mulgt0d

Step	Hyp	Ref	Expression
1		ltd.1	. 2 |- ( ph -> A e. RR )
2		mulgt0d.3	. 2 |- ( ph -> 0 < A )
3		ltd.2	. 2 |- ( ph -> B e. RR )
4		mulgt0d.4	. 2 |- ( ph -> 0 < B )
5		mulgt0 8390	. 2 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A x. B ) )
6	1, 2, 3, 4, 5	syl22anc 1279	1 |- ( ph -> 0 < ( A x. B ) )

Syntax hints:   -> wi 4   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   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:  ltmul1a  8909  mulge0  8937  recgt0  9170  prodgt0gt0  9171  prodge0  9174  modqmulnn  10757  modqdi  10807  cos12dec  12513  tangtx  15862