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Theorem addltmulALT 23042
Description: A proof readability experiment for addltmul 9963. (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 447 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  <  A )
2 2re 9831 . . . . . . . 8  |-  2  e.  RR
32a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
2  e.  RR )
4 simpl 443 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  ->  A  e.  RR )
5 1re 8853 . . . . . . . 8  |-  1  e.  RR
65a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  e.  RR )
7 ltsub1 9286 . . . . . . 7  |-  ( ( 2  e.  RR  /\  A  e.  RR  /\  1  e.  RR )  ->  (
2  <  A  <->  ( 2  -  1 )  < 
( A  -  1 ) ) )
83, 4, 6, 7syl3anc 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  ( 2  -  1 )  <  ( A  - 
1 ) ) )
9 2cn 9832 . . . . . . . . 9  |-  2  e.  CC
10 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
11 df-2 9820 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
1211eqcomi 2300 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
139, 10, 10, 12subaddrii 9151 . . . . . . . 8  |-  ( 2  -  1 )  =  1
1413breq1i 4046 . . . . . . 7  |-  ( ( 2  -  1 )  <  ( A  - 
1 )  <->  1  <  ( A  -  1 ) )
1514a1i 10 . . . . . 6  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( ( 2  -  1 )  <  ( A  -  1 )  <->  1  <  ( A  -  1 ) ) )
168, 15bitrd 244 . . . . 5  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
( 2  <  A  <->  1  <  ( A  - 
1 ) ) )
171, 16mpbid 201 . . . 4  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  <  ( A  -  1 ) )
18 simpr 447 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  <  B )
192a1i 10 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
2  e.  RR )
20 simpl 443 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  ->  B  e.  RR )
215a1i 10 . . . . . . 7  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  e.  RR )
22 ltsub1 9286 . . . . . . 7  |-  ( ( 2  e.  RR  /\  B  e.  RR  /\  1  e.  RR )  ->  (
2  <  B  <->  ( 2  -  1 )  < 
( B  -  1 ) ) )
2319, 20, 21, 22syl3anc 1182 . . . . . 6  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  ( 2  -  1 )  <  ( B  - 
1 ) ) )
2413breq1i 4046 . . . . . . 7  |-  ( ( 2  -  1 )  <  ( B  - 
1 )  <->  1  <  ( B  -  1 ) )
2524a1i 10 . . . . . 6  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( ( 2  -  1 )  <  ( B  -  1 )  <->  1  <  ( B  -  1 ) ) )
2623, 25bitrd 244 . . . . 5  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
( 2  <  B  <->  1  <  ( B  - 
1 ) ) )
2718, 26mpbid 201 . . . 4  |-  ( ( B  e.  RR  /\  2  <  B )  -> 
1  <  ( B  -  1 ) )
2817, 27anim12i 549 . . 3  |-  ( ( ( A  e.  RR  /\  2  <  A )  /\  ( B  e.  RR  /\  2  < 
B ) )  -> 
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) ) )
2928an4s 799 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
1  <  ( A  -  1 )  /\  1  <  ( B  - 
1 ) ) )
30 peano2rem 9129 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
31 peano2rem 9129 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  -  1 )  e.  RR )
3230, 31anim12i 549 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR ) )
3332anim1i 551 . . . . . 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 9631 . . . . . 6  |-  ( ( ( ( A  - 
1 )  e.  RR  /\  ( B  -  1 )  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
3533, 34syl 15 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) ) )  -> 
1  <  ( ( A  -  1 )  x.  ( B  - 
1 ) ) )
3635ex 423 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 1  < 
( A  -  1 )  /\  1  < 
( B  -  1 ) )  ->  1  <  ( ( A  - 
1 )  x.  ( B  -  1 ) ) ) )
3736adantr 451 . . 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 8843 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  e.  CC )
3910a1i 10 . . . . . . . . 9  |-  ( A  e.  RR  ->  1  e.  CC )
4038, 39jca 518 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  e.  CC  /\  1  e.  CC ) )
41 recn 8843 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  e.  CC )
4210a1i 10 . . . . . . . . 9  |-  ( B  e.  RR  ->  1  e.  CC )
4341, 42jca 518 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  e.  CC  /\  1  e.  CC ) )
4440, 43anim12i 549 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  e.  CC  /\  1  e.  CC )  /\  ( B  e.  CC  /\  1  e.  CC ) ) )
45 mulsub 9238 . . . . . . 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 15 . . . . . 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 4051 . . . . 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 198 . . . 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 451 . . 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 8856 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
51 eqcom 2298 . . . . . . . . . 10  |-  ( ( 1  x.  1 )  =  1  <->  1  =  ( 1  x.  1 ) )
5251biimpi 186 . . . . . . . . 9  |-  ( ( 1  x.  1 )  =  1  ->  1  =  ( 1  x.  1 ) )
5350, 52mp1i 11 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  1  =  ( 1  x.  1 ) )
5453oveq2d 5890 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  =  ( ( A  x.  B )  +  ( 1  x.  1 ) ) )
55 mulid1 8851 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
56 eqcom 2298 . . . . . . . . . . . 12  |-  ( ( A  x.  1 )  =  A  <->  A  =  ( A  x.  1
) )
5756biimpi 186 . . . . . . . . . . 11  |-  ( ( A  x.  1 )  =  A  ->  A  =  ( A  x.  1 ) )
5855, 57syl 15 . . . . . . . . . 10  |-  ( A  e.  CC  ->  A  =  ( A  x.  1 ) )
5938, 58syl 15 . . . . . . . . 9  |-  ( A  e.  RR  ->  A  =  ( A  x.  1 ) )
6059adantr 451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  A  =  ( A  x.  1 ) )
61 mulid1 8851 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
6241, 61syl 15 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
63 eqcom 2298 . . . . . . . . . . 11  |-  ( ( B  x.  1 )  =  B  <->  B  =  ( B  x.  1
) )
6463biimpi 186 . . . . . . . . . 10  |-  ( ( B  x.  1 )  =  B  ->  B  =  ( B  x.  1 ) )
6562, 64syl 15 . . . . . . . . 9  |-  ( B  e.  RR  ->  B  =  ( B  x.  1 ) )
6665adantl 452 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  B  =  ( B  x.  1 ) )
6760, 66oveq12d 5892 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )
6854, 67oveq12d 5892 . . . . . 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 4051 . . . . 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 8836 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
715a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  1  e.  RR )
72 remulcl 8838 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
73 readdcl 8836 . . . . . . . 8  |-  ( ( ( A  x.  B
)  e.  RR  /\  1  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
7472, 71, 73syl2anc 642 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  e.  RR )
75 ltaddsub2 9265 . . . . . . 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 1182 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <->  1  <  ( ( ( A  x.  B
)  +  1 )  -  ( A  +  B ) ) ) )
77 ltadd1 9257 . . . . . . . . 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 1182 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  B )  <  ( A  x.  B )  <->  ( ( A  +  B
)  +  1 )  <  ( ( A  x.  B )  +  1 ) ) )
7978bicomd 192 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  <-> 
( A  +  B
)  <  ( A  x.  B ) ) )
8079biimpd 198 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( ( A  +  B )  +  1 )  <  (
( A  x.  B
)  +  1 )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8176, 80sylbird 226 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( 1  <  (
( ( A  x.  B )  +  1 )  -  ( A  +  B ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8269, 81sylbird 226 . . . 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 451 . . 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 51 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 2  < 
A  /\  2  <  B ) )  ->  (
( 1  <  ( A  -  1 )  /\  1  <  ( B  -  1 ) )  ->  ( A  +  B )  <  ( A  x.  B )
) )
8529, 84mpd 14 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   2c2 9811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-2 9820
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