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Theorem addltmulALT 23954
Description: A proof readability experiment for addltmul 10208. (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 449 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  <  A )
2 2re 10074 . . . . . . . 8  |-  2  e.  RR
32a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  e.  RR )
4 simpl 445 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  ->  A  e.  RR )
5 1re 9095 . . . . . . . 8  |-  1  e.  RR
65a1i 11 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  e.  RR )
7 ltsub1 9529 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
83, 4, 6, 7syl3anc 1185 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  ( 2  -  1 )  <  ( A  - 
1 ) ) )
9 2cn 10075 . . . . . . . . 9  |-  2  e.  CC
10 ax-1cn 9053 . . . . . . . . 9  |-  1  e.  CC
11 df-2 10063 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
1211eqcomi 2442 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
139, 10, 10, 12subaddrii 9394 . . . . . . . 8  |-  ( 2  -  1 )  =  1
1413breq1i 4222 . . . . . . 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 246 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  1  <  ( A  - 
1 ) ) )
171, 16mpbid 203 . . . 4  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  <  ( A  -  1 ) )
18 simpr 449 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  <  B )
192a1i 11 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  e.  RR )
20 simpl 445 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  ->  B  e.  RR )
215a1i 11 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  e.  RR )
22 ltsub1 9529 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
2319, 20, 21, 22syl3anc 1185 . . . . . 6  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  ( 2  -  1 )  <  ( B  - 
1 ) ) )
2413breq1i 4222 . . . . . . 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 246 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  1  <  ( B  - 
1 ) ) )
2718, 26mpbid 203 . . . 4  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  <  ( B  -  1 ) )
2817, 27anim12i 551 . . 3  |-  ( ( ( A  e.  RR  /\  2  <  A )  /\  ( B  e.  RR  /\  2  < 
B ) )  -> 
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) ) )
2928an4s 801 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
1  <  ( A  -  1 )  /\  1  <  ( B  - 
1 ) ) )
30 peano2rem 9372 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
31 peano2rem 9372 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
3230, 31anim12i 551 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR ) )
3332anim1i 553 . . . . . 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 9874 . . . . . 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 425 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
3736adantr 453 . . 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 9085 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
3910a1i 11 . . . . . . . . 9  |-  ( A  e.  RR  ->  1  e.  CC )
4038, 39jca 520 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  e.  CC  /\  1  e.  CC ) )
41 recn 9085 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
4210a1i 11 . . . . . . . . 9  |-  ( B  e.  RR  ->  1  e.  CC )
4341, 42jca 520 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  e.  CC  /\  1  e.  CC ) )
4440, 43anim12i 551 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  e.  CC  /\  1  e.  CC )  /\  ( B  e.  CC  /\  1  e.  CC ) ) )
45 mulsub 9481 . . . . . . 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 4227 . . . . 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 200 . . . 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 453 . . 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 9098 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
51 eqcom 2440 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  =  1  <->  1  =  ( 1  x.  1 ) )
5251biimpi 188 . . . . . . . . 9  |-  ( ( 1  x.  1 )  =  1  ->  1  =  ( 1  x.  1 ) )
5350, 52mp1i 12 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  1  =  ( 1  x.  1 ) )
5453oveq2d 6100 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  =  ( ( A  x.  B )  +  ( 1  x.  1 ) ) )
55 mulid1 9093 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
56 eqcom 2440 . . . . . . . . . . . 12  |-  ( ( A  x.  1 )  =  A  <->  A  =  ( A  x.  1
) )
5756biimpi 188 . . . . . . . . . . 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 453 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =  ( A  x.  1 ) )
61 mulid1 9093 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
6241, 61syl 16 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
63 eqcom 2440 . . . . . . . . . . 11  |-  ( ( B  x.  1 )  =  B  <->  B  =  ( B  x.  1
) )
6463biimpi 188 . . . . . . . . . 10  |-  ( ( B  x.  1 )  =  B  ->  B  =  ( B  x.  1 ) )
6562, 64syl 16 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  =  ( B  x.  1 ) )
6665adantl 454 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =  ( B  x.  1 ) )
6760, 66oveq12d 6102 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )
6854, 67oveq12d 6102 . . . . . 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 4227 . . . . 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 9078 . . . . . . 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 9080 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
73 readdcl 9078 . . . . . . . 8  |-  ( ( ( A  x.  B
)  e.  RR  /\  1  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
7472, 71, 73syl2anc 644 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
75 ltaddsub2 9508 . . . . . . 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 1185 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <->  1  <  ( ( ( A  x.  B
)  +  1 )  -  ( A  +  B ) ) ) )
77 ltadd1 9500 . . . . . . . . 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 1185 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <  ( A  x.  B )  <->  ( ( A  +  B
)  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
7978bicomd 194 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
8079biimpd 200 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8176, 80sylbird 228 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  1 )  -  ( A  +  B ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8269, 81sylbird 228 . . . 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 453 . . 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 54 . 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 178    /\ wa 360    = wceq 1653    e. wcel 1726   class class class wbr 4215  (class class class)co 6084   CCcc 8993   RRcr 8994   1c1 8996    + caddc 8998    x. cmul 9000    < clt 9125    - cmin 9296   2c2 10054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-2 10063
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