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Theorem addltmul 8334
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 8176 . . . . . . 7  |-  2  e.  RR
2 1re 7180 . . . . . . 7  |-  1  e.  RR
3 ltsub1 7629 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
41, 2, 3mp3an13 1260 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
5 2m1e1 8223 . . . . . . 7  |-  ( 2  -  1 )  =  1
65breq1i 3800 . . . . . 6  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
74, 6syl6bb 194 . . . . 5  |-  ( A  e.  RR  ->  (
2  <  A  <->  1  <  ( A  -  1 ) ) )
8 ltsub1 7629 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
91, 2, 8mp3an13 1260 . . . . . 6  |-  ( B  e.  RR  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
105breq1i 3800 . . . . . 6  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
119, 10syl6bb 194 . . . . 5  |-  ( B  e.  RR  ->  (
2  <  B  <->  1  <  ( B  -  1 ) ) )
127, 11bi2anan9 571 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  <->  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) ) )
13 peano2rem 7442 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
14 peano2rem 7442 . . . . 5  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
15 mulgt1 8008 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
1615ex 113 . . . . 5  |-  ( ( ( A  -  1 )  e.  RR  /\  ( B  -  1
)  e.  RR )  ->  ( ( 1  <  ( A  - 
1 )  /\  1  <  ( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1713, 14, 16syl2an 283 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1812, 17sylbid 148 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  1  <  ( ( A  -  1 )  x.  ( B  -  1 ) ) ) )
19 recn 7168 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
20 recn 7168 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
21 ax-1cn 7131 . . . . . . 7  |-  1  e.  CC
22 mulsub 7572 . . . . . . . 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 ) ) ) )
2321, 22mpanl2 426 . . . . . . 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 ) ) ) )
2421, 23mpanr2 429 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
2519, 20, 24syl2an 283 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
2625breq2d 3805 . . . 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 ) ) ) ) )
27 1t1e1 8251 . . . . . . 7  |-  ( 1  x.  1 )  =  1
2827oveq2i 5554 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
2928breq2i 3801 . . . . 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 ) )
30 remulcl 7163 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
312, 30mpan2 416 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  e.  RR )
32 remulcl 7163 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  x.  1 )  e.  RR )
332, 32mpan2 416 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  e.  RR )
34 readdcl 7161 . . . . . . 7  |-  ( ( ( A  x.  1 )  e.  RR  /\  ( B  x.  1
)  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
3531, 33, 34syl2an 283 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
36 remulcl 7163 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
372, 2remulcli 7195 . . . . . . 7  |-  ( 1  x.  1 )  e.  RR
38 readdcl 7161 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( 1  x.  1 )  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
3936, 37, 38sylancl 404 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
40 ltaddsub2 7608 . . . . . . 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 ) ) ) ) )
412, 40mp3an2 1257 . . . . . 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 ) ) ) ) )
4235, 39, 41syl2anc 403 . . . . 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 ) ) ) ) )
4329, 42syl5rbbr 193 . . . 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 ) ) )
44 ltadd1 7600 . . . . . . 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 ) ) )
452, 44mp3an3 1258 . . . . . 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 ) ) )
4635, 36, 45syl2anc 403 . . . . 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 ) ) )
47 ax-1rid 7145 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
48 ax-1rid 7145 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
4947, 48oveqan12d 5562 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
5049breq1d 3803 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( A  +  B )  <  ( A  x.  B ) ) )
5146, 50bitr3d 188 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  +  1 )  <  (
( A  x.  B
)  +  1 )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5226, 43, 513bitrd 212 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( A  -  1 )  x.  ( B  -  1 ) )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5318, 52sylibd 147 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  ( A  +  B )  <  ( A  x.  B )
) )
5453imp 122 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    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   CCcc 7041   RRcr 7042   1c1 7044    + caddc 7046    x. cmul 7048    < clt 7215    - cmin 7346   2c2 8156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-1re 7132  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-mulrcl 7137  ax-addcom 7138  ax-mulcom 7139  ax-addass 7140  ax-mulass 7141  ax-distr 7142  ax-i2m1 7143  ax-0lt1 7144  ax-1rid 7145  ax-0id 7146  ax-rnegex 7147  ax-precex 7148  ax-cnre 7149  ax-pre-lttrn 7152  ax-pre-ltadd 7154  ax-pre-mulgt0 7155
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349  df-2 8165
This theorem is referenced by: (None)
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