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Theorem addltmulALT 23026
Description: A proof readability experiment for addltmul 9947. (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 9815 . . . . . . . 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 8837 . . . . . . . 8  |-  1  e.  RR
65a1i 10 . . . . . . 7  |-  ( ( A  e.  RR  /\  2  <  A )  -> 
1  e.  RR )
7 ltsub1 9270 . . . . . . 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 9816 . . . . . . . . 9  |-  2  e.  CC
10 ax-1cn 8795 . . . . . . . . 9  |-  1  e.  CC
11 df-2 9804 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
1211eqcomi 2287 . . . . . . . . 9  |-  ( 1  +  1 )  =  2
139, 10, 10, 12subaddrii 9135 . . . . . . . 8  |-  ( 2  -  1 )  =  1
1413breq1i 4030 . . . . . . 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 9270 . . . . . . 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 4030 . . . . . . 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 9113 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  -  1 )  e.  RR )
31 peano2rem 9113 . . . . . . . 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 9615 . . . . . 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 8827 . . . . . . . . 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 8827 . . . . . . . . 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 9222 . . . . . . 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 4035 . . . . 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 8840 . . . . . . . . 9  |-  ( 1  x.  1 )  =  1
51 eqcom 2285 . . . . . . . . . 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 5874 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  B )  +  1 )  =  ( ( A  x.  B )  +  ( 1  x.  1 ) ) )
55 mulid1 8835 . . . . . . . . . . 11  |-  ( A  e.  CC  ->  ( A  x.  1 )  =  A )
56 eqcom 2285 . . . . . . . . . . . 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 8835 . . . . . . . . . . 11  |-  ( B  e.  CC  ->  ( B  x.  1 )  =  B )
6241, 61syl 15 . . . . . . . . . 10  |-  ( B  e.  RR  ->  ( B  x.  1 )  =  B )
63 eqcom 2285 . . . . . . . . . . 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 5876 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( ( A  x.  1 )  +  ( B  x.  1 ) ) )
6854, 67oveq12d 5876 . . . . . 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 4035 . . . . 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 8820 . . . . . . 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 8822 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B
)  e.  RR )
73 readdcl 8820 . . . . . . . 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 9249 . . . . . . 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 9241 . . . . . . . . 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 1623    e. wcel 1684   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037   2c2 9795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-2 9804
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