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Theorem addltmul 10159
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 10025 . . . . . . 7  |-  2  e.  RR
2 1re 9046 . . . . . . 7  |-  1  e.  RR
3 ltsub1 9480 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
41, 2, 3mp3an13 1270 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
5 2m1e1 10051 . . . . . . 7  |-  ( 2  -  1 )  =  1
65breq1i 4179 . . . . . 6  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
74, 6syl6bb 253 . . . . 5  |-  ( A  e.  RR  ->  (
2  <  A  <->  1  <  ( A  -  1 ) ) )
8 ltsub1 9480 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
91, 2, 8mp3an13 1270 . . . . . 6  |-  ( B  e.  RR  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
105breq1i 4179 . . . . . 6  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
119, 10syl6bb 253 . . . . 5  |-  ( B  e.  RR  ->  (
2  <  B  <->  1  <  ( B  -  1 ) ) )
127, 11bi2anan9 844 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  <->  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) ) )
13 peano2rem 9323 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
14 peano2rem 9323 . . . . 5  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
15 mulgt1 9825 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
1615ex 424 . . . . 5  |-  ( ( ( A  -  1 )  e.  RR  /\  ( B  -  1
)  e.  RR )  ->  ( ( 1  <  ( A  - 
1 )  /\  1  <  ( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1713, 14, 16syl2an 464 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
1812, 17sylbid 207 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  1  <  ( ( A  -  1 )  x.  ( B  -  1 ) ) ) )
19 recn 9036 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
20 recn 9036 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
21 ax-1cn 9004 . . . . . . 7  |-  1  e.  CC
22 mulsub 9432 . . . . . . . 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 663 . . . . . . 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 666 . . . . . 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 464 . . . . 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 4184 . . . 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 10082 . . . . . . 7  |-  ( 1  x.  1 )  =  1
2827oveq2i 6051 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
2928breq2i 4180 . . . . 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 9031 . . . . . . . 8  |-  ( ( A  e.  RR  /\  1  e.  RR )  ->  ( A  x.  1 )  e.  RR )
312, 30mpan2 653 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  e.  RR )
32 remulcl 9031 . . . . . . . 8  |-  ( ( B  e.  RR  /\  1  e.  RR )  ->  ( B  x.  1 )  e.  RR )
332, 32mpan2 653 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  e.  RR )
34 readdcl 9029 . . . . . . 7  |-  ( ( ( A  x.  1 )  e.  RR  /\  ( B  x.  1
)  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
3531, 33, 34syl2an 464 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  e.  RR )
36 remulcl 9031 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
372, 2remulcli 9060 . . . . . . 7  |-  ( 1  x.  1 )  e.  RR
38 readdcl 9029 . . . . . . 7  |-  ( ( ( A  x.  B
)  e.  RR  /\  ( 1  x.  1 )  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
3936, 37, 38sylancl 644 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  ( 1  x.  1 ) )  e.  RR )
40 ltaddsub2 9459 . . . . . . 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 1267 . . . . . 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 643 . . . . 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 252 . . . 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 9451 . . . . . . 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 1268 . . . . . 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 643 . . . . 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 9016 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
48 ax-1rid 9016 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
4947, 48oveqan12d 6059 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
5049breq1d 4182 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  1 )  +  ( B  x.  1 ) )  <  ( A  x.  B )  <->  ( A  +  B )  <  ( A  x.  B ) ) )
5146, 50bitr3d 247 . . . 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 271 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( A  -  1 )  x.  ( B  -  1 ) )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
5318, 52sylibd 206 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  ->  ( A  +  B )  <  ( A  x.  B )
) )
5453imp 419 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   CCcc 8944   RRcr 8945   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247   2c2 10005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-2 10014
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