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Theorem mulgt1 10191
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 457 . . . . 5  |-  ( ( 1  <  A  /\  1  <  B )  -> 
1  <  A )
21a1i 11 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  A ) )
3 0lt1 9865 . . . . . . . . 9  |-  0  <  1
4 0re 9389 . . . . . . . . . 10  |-  0  e.  RR
5 1re 9388 . . . . . . . . . 10  |-  1  e.  RR
6 lttr 9454 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
74, 5, 6mp3an12 1304 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
83, 7mpani 676 . . . . . . . 8  |-  ( A  e.  RR  ->  (
1  <  A  ->  0  <  A ) )
98adantr 465 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  0  <  A ) )
10 ltmul2 10183 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  <->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1110biimpd 207 . . . . . . . . . 10  |-  ( ( 1  e.  RR  /\  B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
125, 11mp3an1 1301 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1312exp32 605 . . . . . . . 8  |-  ( B  e.  RR  ->  ( A  e.  RR  ->  ( 0  <  A  -> 
( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) ) )
1413impcom 430 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 0  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
159, 14syld 44 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) ) )
1615impd 431 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( A  x.  1 )  <  ( A  x.  B )
) )
17 ax-1rid 9355 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
1817adantr 465 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  1 )  =  A )
1918breq1d 4305 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  <  ( A  x.  B )  <->  A  <  ( A  x.  B ) ) )
2016, 19sylibd 214 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  A  <  ( A  x.  B ) ) )
212, 20jcad 533 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  ( 1  <  A  /\  A  <  ( A  x.  B
) ) ) )
22 remulcl 9370 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
23 lttr 9454 . . . . 5  |-  ( ( 1  e.  RR  /\  A  e.  RR  /\  ( A  x.  B )  e.  RR )  ->  (
( 1  <  A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B )
) )
245, 23mp3an1 1301 . . . 4  |-  ( ( A  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( ( 1  <  A  /\  A  <  ( A  x.  B
) )  ->  1  <  ( A  x.  B
) ) )
2522, 24syldan 470 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B ) ) )
2621, 25syld 44 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
A  /\  1  <  B )  ->  1  <  ( A  x.  B ) ) )
2726imp 429 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
A  /\  1  <  B ) )  ->  1  <  ( A  x.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4295  (class class class)co 6094   RRcr 9284   0cc0 9285   1c1 9286    x. cmul 9290    < clt 9421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601
This theorem is referenced by:  mulgt1d  10272  addltmul  10563  uz2mulcl  10935  addltmulALT  25853
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