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Theorem mulgt1 9615
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 443 . . . . 5  |-  ( ( 1  <  A  /\  1  <  B )  -> 
1  <  A )
21a1i 10 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  A ) )
3 0lt1 9296 . . . . . . . . 9  |-  0  <  1
4 0re 8838 . . . . . . . . . 10  |-  0  e.  RR
5 1re 8837 . . . . . . . . . 10  |-  1  e.  RR
6 lttr 8899 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
74, 5, 6mp3an12 1267 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
83, 7mpani 657 . . . . . . . 8  |-  ( A  e.  RR  ->  (
1  <  A  ->  0  <  A ) )
98adantr 451 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  0  <  A ) )
10 ltmul2 9607 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  <->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1110biimpd 198 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
125, 11mp3an1 1264 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1312exp32 588 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  e.  RR  ->  ( 0  <  A  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) ) )
1413impcom 419 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
159, 14syld 40 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
1615imp3a 420 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( A  x.  1 )  <  ( A  x.  B )
) )
17 ax-1rid 8807 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
1817adantr 451 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  1 )  =  A )
1918breq1d 4033 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  <  ( A  x.  B )  <->  A  <  ( A  x.  B ) ) )
2016, 19sylibd 205 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  A  <  ( A  x.  B ) ) )
212, 20jcad 519 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( 1  <  A  /\  A  <  ( A  x.  B
) ) ) )
22 remulcl 8822 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
23 lttr 8899 . . . . 5  |-  ( ( 1  e.  RR  /\  A  e.  RR  /\  ( A  x.  B )  e.  RR )  ->  (
( 1  <  A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B )
) )
245, 23mp3an1 1264 . . . 4  |-  ( ( A  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( ( 1  <  A  /\  A  <  ( A  x.  B
) )  ->  1  <  ( A  x.  B
) ) )
2522, 24syldan 456 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B ) ) )
2621, 25syld 40 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  ( A  x.  B ) ) )
2726imp 418 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    x. cmul 8742    < clt 8867
This theorem is referenced by:  mulgt1d  9693  addltmul  9947  uz2mulcl  10295  addltmulALT  23026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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