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Theorem ltdivp1i 10251
Description: Less-than and division relation. (Lemma for computing upper bounds of products. The "+ 1" prevents division by zero.) (Contributed by NM, 17-Sep-2005.)
Hypotheses
Ref Expression
ltplus1.1  |-  A  e.  RR
prodgt0.2  |-  B  e.  RR
ltmul1.3  |-  C  e.  RR
Assertion
Ref Expression
ltdivp1i  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <  B )

Proof of Theorem ltdivp1i
StepHypRef Expression
1 ltplus1.1 . . . 4  |-  A  e.  RR
2 ltmul1.3 . . . . 5  |-  C  e.  RR
3 1re 9377 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9391 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10228 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9489 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10176 . . . . . 6  |-  ( ( ( C  e.  RR  /\  ( C  +  1 )  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  /\  C  <_  ( C  +  1 ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
86, 7mpan2 671 . . . . 5  |-  ( ( C  e.  RR  /\  ( C  +  1
)  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
92, 4, 8mp3an12 1304 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) ) )
101, 9mpan 670 . . 3  |-  ( 0  <_  A  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
11103ad2ant1 1009 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 9378 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9493 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 671 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9867 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10089 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 16 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 ltmul1 10171 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  ( B  /  ( C  +  1 ) )  e.  RR  /\  ( ( C  + 
1 )  e.  RR  /\  0  <  ( C  +  1 ) ) )  ->  ( A  <  ( B  /  ( C  +  1 ) )  <->  ( A  x.  ( C  +  1
) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
201, 19mp3an1 1301 . . . . . . . . 9  |-  ( ( ( B  /  ( C  +  1 ) )  e.  RR  /\  ( ( C  + 
1 )  e.  RR  /\  0  <  ( C  +  1 ) ) )  ->  ( A  <  ( B  /  ( C  +  1 ) )  <->  ( A  x.  ( C  +  1
) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
214, 20mpanr1 683 . . . . . . . 8  |-  ( ( ( B  /  ( C  +  1 ) )  e.  RR  /\  0  <  ( C  + 
1 ) )  -> 
( A  <  ( B  /  ( C  + 
1 ) )  <->  ( A  x.  ( C  +  1 ) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2218, 21mpancom 669 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2322biimpd 207 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2414, 23syl 16 . . . . 5  |-  ( 0  <_  C  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2524imp 429 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  +  1 ) )  x.  ( C  + 
1 ) ) )
2616recni 9390 . . . . . . 7  |-  B  e.  CC
274recni 9390 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2826, 27divcan1zi 10059 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
2914, 15, 283syl 20 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3029adantr 465 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3125, 30breqtrd 4311 . . 3  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  B )
32313adant1 1006 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <  B )
331, 2remulcli 9392 . . 3  |-  ( A  x.  C )  e.  RR
341, 4remulcli 9392 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3533, 34, 16lelttri 9493 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <  B )  -> 
( A  x.  C
)  <  B )
3611, 32, 35syl2anc 661 1  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   class class class wbr 4287  (class class class)co 6086   RRcr 9273   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279    < clt 9410    <_ cle 9411    / cdiv 9985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986
This theorem is referenced by: (None)
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