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Theorem addltmul 9157
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 8991 . . . . . . 7  |-  2  e.  RR
2 1re 7958 . . . . . . 7  |-  1  e.  RR
3 ltsub1 8417 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
41, 2, 3mp3an13 1328 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
5 2m1e1 9039 . . . . . . 7  |-  ( 2  -  1 )  =  1
65breq1i 4012 . . . . . 6  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
74, 6bitrdi 196 . . . . 5  |-  ( A  e.  RR  ->  (
2  <  A  <->  1  <  ( A  -  1 ) ) )
8 ltsub1 8417 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
91, 2, 8mp3an13 1328 . . . . . 6  |-  ( B  e.  RR  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
105breq1i 4012 . . . . . 6  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
119, 10bitrdi 196 . . . . 5  |-  ( B  e.  RR  ->  (
2  <  B  <->  1  <  ( B  -  1 ) ) )
127, 11bi2anan9 606 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  <->  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) ) )
13 peano2rem 8226 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
14 peano2rem 8226 . . . . 5  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
15 mulgt1 8822 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
1615ex 115 . . . . 5  |-  ( ( ( A  -  1 )  e.  RR  /\  ( B  -  1
)  e.  RR )  ->  ( ( 1  <  ( A  - 
1 )  /\  1  <  ( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1713, 14, 16syl2an 289 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1812, 17sylbid 150 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  1  <  ( ( A  -  1 )  x.  ( B  -  1 ) ) ) )
19 recn 7946 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
20 recn 7946 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
21 ax-1cn 7906 . . . . . . 7  |-  1  e.  CC
22 mulsub 8360 . . . . . . . 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 435 . . . . . . 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 438 . . . . . 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 289 . . . . 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 4017 . . . 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 remulcl 7941 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
282, 27mpan2 425 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  e.  RR )
29 remulcl 7941 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  x.  1 )  e.  RR )
302, 29mpan2 425 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  e.  RR )
31 readdcl 7939 . . . . . . 7  |-  ( ( ( A  x.  1 )  e.  RR  /\  ( B  x.  1
)  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
3228, 30, 31syl2an 289 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
33 remulcl 7941 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
342, 2remulcli 7973 . . . . . . 7  |-  ( 1  x.  1 )  e.  RR
35 readdcl 7939 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( 1  x.  1 )  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
3633, 34, 35sylancl 413 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
37 ltaddsub2 8396 . . . . . . 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 ) ) ) ) )
382, 37mp3an2 1325 . . . . . 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 ) ) ) ) )
3932, 36, 38syl2anc 411 . . . . 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 ) ) ) ) )
40 1t1e1 9073 . . . . . . 7  |-  ( 1  x.  1 )  =  1
4140oveq2i 5888 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
4241breq2i 4013 . . . . 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 ) )
4339, 42bitr3di 195 . . . 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 8388 . . . . . . 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 1326 . . . . . 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 ) ) )
4632, 33, 45syl2anc 411 . . . . 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 7920 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
48 ax-1rid 7920 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
4947, 48oveqan12d 5896 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
5049breq1d 4015 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( A  +  B )  <  ( A  x.  B ) ) )
5146, 50bitr3d 190 . . . 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 214 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( A  -  1 )  x.  ( B  -  1 ) )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5318, 52sylibd 149 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  ( A  +  B )  <  ( A  x.  B )
) )
5453imp 124 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 104    <-> wb 105    = wceq 1353    e. wcel 2148   class class class wbr 4005  (class class class)co 5877   CCcc 7811   RRcr 7812   1c1 7814    + caddc 7816    x. cmul 7818    < clt 7994    - cmin 8130   2c2 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-lttrn 7927  ax-pre-ltadd 7929  ax-pre-mulgt0 7930
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-sub 8132  df-neg 8133  df-2 8980
This theorem is referenced by: (None)
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