ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  mulgt1 Unicode version

Theorem mulgt1 8779
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 8046 . . . . . . . . 9  |-  0  <  1
4 0re 7920 . . . . . . . . . 10  |-  0  e.  RR
5 1re 7919 . . . . . . . . . 10  |-  1  e.  RR
6 lttr 7993 . . . . . . . . . 10  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  A  e.  RR )  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
74, 5, 6mp3an12 1322 . . . . . . . . 9  |-  ( A  e.  RR  ->  (
( 0  <  1  /\  1  <  A )  ->  0  <  A
) )
83, 7mpani 428 . . . . . . . 8  |-  ( A  e.  RR  ->  (
1  <  A  ->  0  <  A ) )
98adantr 274 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  A  ->  0  <  A ) )
10 ltmul2 8772 . . . . . . . . . . 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 1319 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  ( A  e.  RR  /\  0  <  A ) )  ->  ( 1  <  B  ->  ( A  x.  1 )  <  ( A  x.  B ) ) )
1312exp32 363 . . . . . . . 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 7881 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
1817adantr 274 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  1 )  =  A )
1918breq1d 3999 . . . . 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 7902 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
23 lttr 7993 . . . . 5  |-  ( ( 1  e.  RR  /\  A  e.  RR  /\  ( A  x.  B )  e.  RR )  ->  (
( 1  <  A  /\  A  <  ( A  x.  B ) )  ->  1  <  ( A  x.  B )
) )
245, 23mp3an1 1319 . . . 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 973    = wceq 1348    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775    x. cmul 7779    < clt 7954
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-lttrn 7888  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-ltxr 7959  df-sub 8092  df-neg 8093
This theorem is referenced by:  mulgt1d  8852  addltmul  9114  uz2mulcl  9567
  Copyright terms: Public domain W3C validator