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Theorem addltmulALT 23790
Description: A proof readability experiment for addltmul 10128. (Contributed by Stefan Allan, 30-Oct-2010.) (Proof modification is discouraged.)
Assertion
Ref Expression
addltmulALT  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  ( A  +  B )  <  ( A  x.  B
) )

Proof of Theorem addltmulALT
StepHypRef Expression
1 simpr 448 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  <  A )
2 2re 9994 . . . . . . . 8  |-  2  e.  RR
32a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  e.  RR )
4 simpl 444 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  ->  A  e.  RR )
5 1re 9016 . . . . . . . 8  |-  1  e.  RR
65a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  e.  RR )
7 ltsub1 9449 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
83, 4, 6, 7syl3anc 1184 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  ( 2  -  1 )  <  ( A  - 
1 ) ) )
9 2cn 9995 . . . . . . . . 9  |-  2  e.  CC
10 ax-1cn 8974 . . . . . . . . 9  |-  1  e.  CC
11 df-2 9983 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
1211eqcomi 2384 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
139, 10, 10, 12subaddrii 9314 . . . . . . . 8  |-  ( 2  -  1 )  =  1
1413breq1i 4153 . . . . . . 7  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
1514a1i 11 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( ( 2  -  1 )  <  ( A  -  1 )  <->  1  <  ( A  -  1 ) ) )
168, 15bitrd 245 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  1  <  ( A  - 
1 ) ) )
171, 16mpbid 202 . . . 4  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  <  ( A  -  1 ) )
18 simpr 448 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  <  B )
192a1i 11 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  e.  RR )
20 simpl 444 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  ->  B  e.  RR )
215a1i 11 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  e.  RR )
22 ltsub1 9449 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
2319, 20, 21, 22syl3anc 1184 . . . . . 6  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  ( 2  -  1 )  <  ( B  - 
1 ) ) )
2413breq1i 4153 . . . . . . 7  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
2524a1i 11 . . . . . 6  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( ( 2  -  1 )  <  ( B  -  1 )  <->  1  <  ( B  -  1 ) ) )
2623, 25bitrd 245 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  1  <  ( B  - 
1 ) ) )
2718, 26mpbid 202 . . . 4  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  <  ( B  -  1 ) )
2817, 27anim12i 550 . . 3  |-  ( ( ( A  e.  RR  /\  2  <  A )  /\  ( B  e.  RR  /\  2  < 
B ) )  -> 
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) ) )
2928an4s 800 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
1  <  ( A  -  1 )  /\  1  <  ( B  - 
1 ) ) )
30 peano2rem 9292 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
31 peano2rem 9292 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
3230, 31anim12i 550 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR ) )
3332anim1i 552 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
( ( ( A  -  1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  (
1  <  ( A  -  1 )  /\  1  <  ( B  - 
1 ) ) ) )
34 mulgt1 9794 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
3533, 34syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
3635ex 424 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
3736adantr 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) )  ->  1  <  ( ( A  -  1 )  x.  ( B  -  1 ) ) ) )
38 recn 9006 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
3910a1i 11 . . . . . . . . 9  |-  ( A  e.  RR  ->  1  e.  CC )
4038, 39jca 519 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  e.  CC  /\  1  e.  CC ) )
41 recn 9006 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
4210a1i 11 . . . . . . . . 9  |-  ( B  e.  RR  ->  1  e.  CC )
4341, 42jca 519 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  e.  CC  /\  1  e.  CC ) )
4440, 43anim12i 550 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  e.  CC  /\  1  e.  CC )  /\  ( B  e.  CC  /\  1  e.  CC ) ) )
45 mulsub 9401 . . . . . . 7  |-  ( ( ( 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 ) ) ) )
4644, 45syl 16 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  x.  ( B  -  1 ) )  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
4746breq2d 4158 . . . . 5  |-  ( ( 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 ) ) ) ) )
4847biimpd 199 . . . 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 ) ) ) ) )
4948adantr 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) )  -> 
1  <  ( (
( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
5010mulid2i 9019 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
51 eqcom 2382 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  =  1  <->  1  =  ( 1  x.  1 ) )
5251biimpi 187 . . . . . . . . 9  |-  ( ( 1  x.  1 )  =  1  ->  1  =  ( 1  x.  1 ) )
5350, 52mp1i 12 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  1  =  ( 1  x.  1 ) )
5453oveq2d 6029 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  =  ( ( A  x.  B )  +  ( 1  x.  1 ) ) )
55 mulid1 9014 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
56 eqcom 2382 . . . . . . . . . . . 12  |-  ( ( A  x.  1 )  =  A  <->  A  =  ( A  x.  1
) )
5756biimpi 187 . . . . . . . . . . 11  |-  ( ( A  x.  1 )  =  A  ->  A  =  ( A  x.  1 ) )
5855, 57syl 16 . . . . . . . . . 10  |-  ( A  e.  CC  ->  A  =  ( A  x.  1 ) )
5938, 58syl 16 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =  ( A  x.  1 ) )
6059adantr 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =  ( A  x.  1 ) )
61 mulid1 9014 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
6241, 61syl 16 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
63 eqcom 2382 . . . . . . . . . . 11  |-  ( ( B  x.  1 )  =  B  <->  B  =  ( B  x.  1
) )
6463biimpi 187 . . . . . . . . . 10  |-  ( ( B  x.  1 )  =  B  ->  B  =  ( B  x.  1 ) )
6562, 64syl 16 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  =  ( B  x.  1 ) )
6665adantl 453 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =  ( B  x.  1 ) )
6760, 66oveq12d 6031 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )
6854, 67oveq12d 6031 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  x.  B )  +  1 )  -  ( A  +  B )
)  =  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) )
6968breq2d 4158 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  1 )  -  ( A  +  B ) )  <->  1  <  ( ( ( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) ) ) )
70 readdcl 8999 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
715a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  1  e.  RR )
72 remulcl 9001 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
73 readdcl 8999 . . . . . . . 8  |-  ( ( ( A  x.  B
)  e.  RR  /\  1  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
7472, 71, 73syl2anc 643 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
75 ltaddsub2 9428 . . . . . . 7  |-  ( ( ( A  +  B
)  e.  RR  /\  1  e.  RR  /\  (
( A  x.  B
)  +  1 )  e.  RR )  -> 
( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <->  1  <  ( ( ( A  x.  B
)  +  1 )  -  ( A  +  B ) ) ) )
7670, 71, 74, 75syl3anc 1184 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <->  1  <  ( ( ( A  x.  B
)  +  1 )  -  ( A  +  B ) ) ) )
77 ltadd1 9420 . . . . . . . . 9  |-  ( ( ( A  +  B
)  e.  RR  /\  ( A  x.  B
)  e.  RR  /\  1  e.  RR )  ->  ( ( A  +  B )  <  ( A  x.  B )  <->  ( ( A  +  B
)  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
7870, 72, 71, 77syl3anc 1184 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <  ( A  x.  B )  <->  ( ( A  +  B
)  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
7978bicomd 193 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
8079biimpd 199 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8176, 80sylbird 227 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  1 )  -  ( A  +  B ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8269, 81sylbird 227 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8382adantr 452 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
1  <  ( (
( A  x.  B
)  +  ( 1  x.  1 ) )  -  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )  -> 
( A  +  B
)  <  ( A  x.  B ) ) )
8437, 49, 833syld 53 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8529, 84mpd 15 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 1717   class class class wbr 4146  (class class class)co 6013   CCcc 8914   RRcr 8915   1c1 8917    + caddc 8919    x. cmul 8921    < clt 9046    - cmin 9216   2c2 9974
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-2 9983
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