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Theorem addltmul 8622
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 8463 . . . . . . 7  |-  2  e.  RR
2 1re 7466 . . . . . . 7  |-  1  e.  RR
3 ltsub1 7915 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
41, 2, 3mp3an13 1264 . . . . . 6  |-  ( A  e.  RR  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
5 2m1e1 8510 . . . . . . 7  |-  ( 2  -  1 )  =  1
65breq1i 3844 . . . . . 6  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
74, 6syl6bb 194 . . . . 5  |-  ( A  e.  RR  ->  (
2  <  A  <->  1  <  ( A  -  1 ) ) )
8 ltsub1 7915 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
91, 2, 8mp3an13 1264 . . . . . 6  |-  ( B  e.  RR  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
105breq1i 3844 . . . . . 6  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
119, 10syl6bb 194 . . . . 5  |-  ( B  e.  RR  ->  (
2  <  B  <->  1  <  ( B  -  1 ) ) )
127, 11bi2anan9 573 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 2  < 
A  /\  2  <  B )  <->  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) ) )
13 peano2rem 7728 . . . . 5  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
14 peano2rem 7728 . . . . 5  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
15 mulgt1 8296 . . . . . 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 7454 . . . . . 6  |-  ( A  e.  RR  ->  A  e.  CC )
20 recn 7454 . . . . . 6  |-  ( B  e.  RR  ->  B  e.  CC )
21 ax-1cn 7417 . . . . . . 7  |-  1  e.  CC
22 mulsub 7858 . . . . . . . 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 3849 . . . 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 8538 . . . . . . 7  |-  ( 1  x.  1 )  =  1
2827oveq2i 5645 . . . . . 6  |-  ( ( A  x.  B )  +  ( 1  x.  1 ) )  =  ( ( A  x.  B )  +  1 )
2928breq2i 3845 . . . . 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 7449 . . . . . . . 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 7449 . . . . . . . 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 7447 . . . . . . 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 7449 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
372, 2remulcli 7481 . . . . . . 7  |-  ( 1  x.  1 )  e.  RR
38 readdcl 7447 . . . . . . 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 7894 . . . . . . 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 1261 . . . . . 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 7886 . . . . . . 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 1262 . . . . . 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 7431 . . . . . . 7  |-  ( A  e.  RR  ->  ( A  x.  1 )  =  A )
48 ax-1rid 7431 . . . . . . 7  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
4947, 48oveqan12d 5653 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  1 )  +  ( B  x.  1 ) )  =  ( A  +  B ) )
5049breq1d 3847 . . . . 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 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   1c1 7330    + caddc 7332    x. cmul 7334    < clt 7501    - cmin 7632   2c2 8444
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-lttrn 7438  ax-pre-ltadd 7440  ax-pre-mulgt0 7441
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635  df-2 8452
This theorem is referenced by: (None)
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