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Theorem axmulgt0 6888
Description: The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. (This restates ax-pre-mulgt0 6800 with ordering on the extended reals.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
axmulgt0  RR  RR  0  <  0  <  0  <  x.

Proof of Theorem axmulgt0
StepHypRef Expression
1 ax-pre-mulgt0 6800 . 2  RR  RR  0  <RR  0  <RR  0  <RR  x.
2 0re 6825 . . . 4  0  RR
3 ltxrlt 6882 . . . 4  0  RR  RR  0  < 
0  <RR
42, 3mpan 400 . . 3  RR 
0  <  0  <RR
5 ltxrlt 6882 . . . 4  0  RR  RR  0  < 
0  <RR
62, 5mpan 400 . . 3  RR 
0  <  0  <RR
74, 6bi2anan9 538 . 2  RR  RR  0  <  0  <  0  <RR  0  <RR
8 remulcl 6807 . . 3  RR  RR  x.  RR
9 ltxrlt 6882 . . 3  0  RR  x.  RR  0  <  x.  0  <RR  x.
102, 8, 9sylancr 393 . 2  RR  RR  0  <  x.  0  <RR  x.
111, 7, 103imtr4d 192 1  RR  RR  0  <  0  <  0  <  x.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wcel 1390   class class class wbr 3755  (class class class)co 5455   RRcr 6710   0cc0 6711    <RR cltrr 6715    x. cmul 6716    < clt 6857
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1re 6777  ax-addrcl 6780  ax-mulrcl 6782  ax-rnegex 6792  ax-pre-mulgt0 6800
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-pnf 6859  df-mnf 6860  df-ltxr 6862
This theorem is referenced by:  mulgt0  6890  mulgt0i  6924
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