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Theorem mulgt1 8589
Description: The product of two numbers greater than 1 is greater than 1. (Contributed by NM, 13-Feb-2005.)
Assertion
Ref Expression
mulgt1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  1  <  ( A  x.  B
) )

Proof of Theorem mulgt1
StepHypRef Expression
1 simpl 108 . . . . 5  |-  ( ( 1  <  A  /\  1  <  B )  -> 
1  <  A )
21a1i 9 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  A ) )
3 0lt1 7857 . . . . . . . . 9  |-  0  <  1
4 0re 7734 . . . . . . . . . 10  |-  0  e.  RR
5 1re 7733 . . . . . . . . . 10  |-  1  e.  RR
6 lttr 7806 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
74, 5, 6mp3an12 1290 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
83, 7mpani 426 . . . . . . . 8  |-  ( A  e.  RR  ->  (
1  <  A  ->  0  <  A ) )
98adantr 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  0  <  A ) )
10 ltmul2 8582 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  <->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1110biimpd 143 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
125, 11mp3an1 1287 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1312exp32 362 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  e.  RR  ->  ( 0  <  A  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) ) )
1413impcom 124 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
159, 14syld 45 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
1615impd 252 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( A  x.  1 )  <  ( A  x.  B )
) )
17 ax-1rid 7695 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
1817adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  1 )  =  A )
1918breq1d 3909 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  <  ( A  x.  B )  <->  A  <  ( A  x.  B ) ) )
2016, 19sylibd 148 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  A  <  ( A  x.  B ) ) )
212, 20jcad 305 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( 1  <  A  /\  A  <  ( A  x.  B
) ) ) )
22 remulcl 7716 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
23 lttr 7806 . . . . 5  |-  ( ( 1  e.  RR  /\  A  e.  RR  /\  ( A  x.  B )  e.  RR )  ->  (
( 1  <  A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B )
) )
245, 23mp3an1 1287 . . . 4  |-  ( ( A  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( ( 1  <  A  /\  A  <  ( A  x.  B
) )  ->  1  <  ( A  x.  B
) ) )
2522, 24syldan 280 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B ) ) )
2621, 25syld 45 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  ( A  x.  B ) ) )
2726imp 123 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  1  <  ( A  x.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 947    = wceq 1316    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   0cc0 7588   1c1 7589    x. cmul 7593    < clt 7768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-lttrn 7702  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904
This theorem is referenced by:  mulgt1d  8662  addltmul  8924  uz2mulcl  9370
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