Theorem axmulgt0 7091

Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 7001 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 7001	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 <RR A /\ 0 <RR B ) -> 0 <RR ( A x. B ) ) )
2		0re 7027	. . . 4 |- 0 e. RR
3		ltxrlt 7085	. . . 4 |- ( ( 0 e. RR /\ A e. RR ) -> ( 0 < A <-> 0 <RR A ) )
4	2, 3	mpan 400	. . 3 |- ( A e. RR -> ( 0 < A <-> 0 <RR A ) )
5		ltxrlt 7085	. . . 4 |- ( ( 0 e. RR /\ B e. RR ) -> ( 0 < B <-> 0 <RR B ) )
6	2, 5	mpan 400	. . 3 |- ( B e. RR -> ( 0 < B <-> 0 <RR B ) )
7	4, 6	bi2anan9 538	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) <-> ( 0 <RR A /\ 0 <RR B ) ) )
8		remulcl 7009	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
9		ltxrlt 7085	. . 3 |- ( ( 0 e. RR /\ ( A x. B ) e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
10	2, 8, 9	sylancr 393	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( A x. B ) <-> 0 <RR ( A x. B ) ) )
11	1, 7, 10	3imtr4d 192	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A /\ 0 < B ) -> 0 < ( A x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888   0cc0 6889   <RR cltrr 6893   x. cmul 6894   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981  ax-mulrcl 6983  ax-rnegex 6993  ax-pre-mulgt0 7001

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065

This theorem is referenced by:  mulgt0  7093  mulgt0i  7127