MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  addltmul Unicode version

Theorem addltmul 9963
Description: Sum is less than product for numbers greater than 2. (Contributed by Stefan Allan, 24-Sep-2010.)
Assertion
Ref Expression
addltmul  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  ( A  +  B )  <  ( A  x.  B
) )

Proof of Theorem addltmul
StepHypRef Expression
1 2re 9831 . . . . . . 7  |-  2  e.  RR
2 1re 8853 . . . . . . 7  |-  1  e.  RR
3 ltsub1 9286 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
41, 2, 3mp3an13 1268 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
5 2cn 9832 . . . . . . . 8  |-  2  e.  CC
6 ax-1cn 8811 . . . . . . . 8  |-  1  e.  CC
7 1p1e2 9856 . . . . . . . 8  |-  ( 1  +  1 )  =  2
85, 6, 6, 7subaddrii 9151 . . . . . . 7  |-  ( 2  -  1 )  =  1
98breq1i 4046 . . . . . 6  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
104, 9syl6bb 252 . . . . 5  |-  ( A  e.  RR  ->  (
2  <  A  <->  1  <  ( A  -  1 ) ) )
11 ltsub1 9286 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
121, 2, 11mp3an13 1268 . . . . . 6  |-  ( B  e.  RR  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
138breq1i 4046 . . . . . 6  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
1412, 13syl6bb 252 . . . . 5  |-  ( B  e.  RR  ->  (
2  <  B  <->  1  <  ( B  -  1 ) ) )
1510, 14bi2anan9 843 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  <->  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) ) )
16 peano2rem 9129 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
17 peano2rem 9129 . . . . 5  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
18 mulgt1 9631 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
1918ex 423 . . . . 5  |-  ( ( ( A  -  1 )  e.  RR  /\  ( B  -  1
)  e.  RR )  ->  ( ( 1  <  ( A  - 
1 )  /\  1  <  ( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
2016, 17, 19syl2an 463 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
2115, 20sylbid 206 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  1  <  ( ( A  -  1 )  x.  ( B  -  1 ) ) ) )
22 recn 8843 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
23 recn 8843 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
24 mulsub 9238 . . . . . . . 8  |-  ( ( ( A  e.  CC  /\  1  e.  CC )  /\  ( B  e.  CC  /\  1  e.  CC ) )  -> 
( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
256, 24mpanl2 662 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( B  e.  CC  /\  1  e.  CC ) )  ->  ( ( A  -  1 )  x.  ( B  - 
1 ) )  =  ( ( ( A  x.  B )  +  ( 1  x.  1 ) )  -  (
( A  x.  1 )  +  ( B  x.  1 ) ) ) )
266, 25mpanr2 665 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
2722, 23, 26syl2an 463 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
2827breq2d 4051 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( A  -  1 )  x.  ( B  -  1 ) )  <->  1  <  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
29 1t1e1 9886 . . . . . . 7  |-  ( 1  x.  1 )  =  1
3029oveq2i 5885 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
3130breq2i 4047 . . . . 5  |-  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  ( ( A  x.  B )  +  ( 1  x.  1 ) )  <->  ( (
( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  < 
( ( A  x.  B )  +  1 ) )
32 remulcl 8838 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
332, 32mpan2 652 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  e.  RR )
34 remulcl 8838 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  x.  1 )  e.  RR )
352, 34mpan2 652 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  e.  RR )
36 readdcl 8836 . . . . . . 7  |-  ( ( ( A  x.  1 )  e.  RR  /\  ( B  x.  1
)  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
3733, 35, 36syl2an 463 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
38 remulcl 8838 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
392, 2remulcli 8867 . . . . . . 7  |-  ( 1  x.  1 )  e.  RR
40 readdcl 8836 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( 1  x.  1 )  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
4138, 39, 40sylancl 643 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
42 ltaddsub2 9265 . . . . . . 7  |-  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR  /\  1  e.  RR  /\  (
( A  x.  B
)  +  ( 1  x.  1 ) )  e.  RR )  -> 
( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  (
( A  x.  B
)  +  ( 1  x.  1 ) )  <->  1  <  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
432, 42mp3an2 1265 . . . . . 6  |-  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR  /\  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )  ->  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  < 
( ( A  x.  B )  +  ( 1  x.  1 ) )  <->  1  <  (
( ( A  x.  B )  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
4437, 41, 43syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  (
( A  x.  B
)  +  ( 1  x.  1 ) )  <->  1  <  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
4531, 44syl5rbbr 251 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )  <-> 
( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
46 ltadd1 9257 . . . . . . 7  |-  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR  /\  ( A  x.  B
)  e.  RR  /\  1  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
472, 46mp3an3 1266 . . . . . 6  |-  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR  /\  ( A  x.  B
)  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  < 
( A  x.  B
)  <->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  (
( A  x.  B
)  +  1 ) ) )
4837, 38, 47syl2anc 642 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
49 ax-1rid 8823 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
50 ax-1rid 8823 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
5149, 50oveqan12d 5893 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
5251breq1d 4049 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( A  +  B )  <  ( A  x.  B ) ) )
5348, 52bitr3d 246 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  (
( A  x.  B
)  +  1 )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5428, 45, 533bitrd 270 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( A  -  1 )  x.  ( B  -  1 ) )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5521, 54sylibd 205 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  ( A  +  B )  <  ( A  x.  B )
) )
5655imp 418 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  ( A  +  B )  <  ( A  x.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   2c2 9811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-2 9820
  Copyright terms: Public domain W3C validator