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Theorem ledivp1i 9682
Description: Less-than-or-equal-to 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
ledivp1i  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  B )

Proof of Theorem ledivp1i
StepHypRef Expression
1 ltplus1.1 . . . 4  |-  A  e.  RR
2 ltmul1.3 . . . . 5  |-  C  e.  RR
3 1re 8837 . . . . . 6  |-  1  e.  RR
42, 3readdcli 8850 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 9660 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 8941 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 9611 . . . . . 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 652 . . . . 5  |-  ( ( C  e.  RR  /\  ( C  +  1
)  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
92, 4, 8mp3an12 1267 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) ) )
101, 9mpan 651 . . 3  |-  ( 0  <_  A  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
11103ad2ant1 976 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 8838 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 8946 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 652 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9308 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 9526 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 15 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 lemul1 9608 . . . . . . . . . . 11  |-  ( ( 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 1264 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) )
2120ex 423 . . . . . . . . 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 ) ) ) ) )
224, 21mpani 657 . . . . . . . 8  |-  ( ( B  /  ( C  +  1 ) )  e.  RR  ->  (
0  <  ( C  +  1 )  -> 
( A  <_  ( B  /  ( C  + 
1 ) )  <->  ( A  x.  ( C  +  1 ) )  <_  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) ) )
2318, 22mpcom 32 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <_  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2423biimpd 198 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2514, 24syl 15 . . . . 5  |-  ( 0  <_  C  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2625imp 418 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) )
2716recni 8849 . . . . . . 7  |-  B  e.  CC
284recni 8849 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2927, 28divcan1zi 9496 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3014, 15, 293syl 18 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3130adantr 451 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3226, 31breqtrd 4047 . . 3  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  B )
33323adant1 973 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <_  B )
341, 2remulcli 8851 . . 3  |-  ( A  x.  C )  e.  RR
351, 4remulcli 8851 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3634, 35, 16letri 8948 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <_  B )  -> 
( A  x.  C
)  <_  B )
3711, 33, 36syl2anc 642 1  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868    / cdiv 9423
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-rmo 2551  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-div 9424
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