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Theorem ltdivp1i 10533
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 9641 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9655 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10510 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9756 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10459 . . . . . 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 675 . . . . 5  |-  ( ( C  e.  RR  /\  ( C  +  1
)  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
92, 4, 8mp3an12 1350 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) ) )
101, 9mpan 674 . . 3  |-  ( 0  <_  A  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
11103ad2ant1 1026 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 9642 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9760 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 675 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 10148 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10372 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 17 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 ltmul1 10454 . . . . . . . . . 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 1347 . . . . . . . . 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 687 . . . . . . . 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 673 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2322biimpd 210 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2414, 23syl 17 . . . . 5  |-  ( 0  <_  C  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2524imp 430 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  +  1 ) )  x.  ( C  + 
1 ) ) )
2616recni 9654 . . . . . . 7  |-  B  e.  CC
274recni 9654 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2826, 27divcan1zi 10342 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
2914, 15, 283syl 18 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3029adantr 466 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3125, 30breqtrd 4450 . . 3  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  B )
32313adant1 1023 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <  B )
331, 2remulcli 9656 . . 3  |-  ( A  x.  C )  e.  RR
341, 4remulcli 9656 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3533, 34, 16lelttri 9760 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <  B )  -> 
( A  x.  C
)  <  B )
3611, 32, 35syl2anc 665 1  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426  (class class class)co 6305   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675    / cdiv 10268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269
This theorem is referenced by: (None)
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