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Theorem ledivp1i 9936
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 9090 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9103 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 9914 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9196 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 9865 . . . . . 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 653 . . . . 5  |-  ( ( C  e.  RR  /\  ( C  +  1
)  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
92, 4, 8mp3an12 1269 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) ) )
101, 9mpan 652 . . 3  |-  ( 0  <_  A  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
11103ad2ant1 978 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 9091 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9200 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 653 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9562 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 9780 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 16 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 lemul1 9862 . . . . . . . . . . 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 1266 . . . . . . . . . 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 424 . . . . . . . . 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 658 . . . . . . . 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 34 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <_  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2423biimpd 199 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2514, 24syl 16 . . . . 5  |-  ( 0  <_  C  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2625imp 419 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) )
2716recni 9102 . . . . . . 7  |-  B  e.  CC
284recni 9102 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2927, 28divcan1zi 9750 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3014, 15, 293syl 19 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3130adantr 452 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3226, 31breqtrd 4236 . . 3  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  B )
33323adant1 975 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <_  B )
341, 2remulcli 9104 . . 3  |-  ( A  x.  C )  e.  RR
351, 4remulcli 9104 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3634, 35, 16letri 9202 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <_  B )  -> 
( A  x.  C
)  <_  B )
3711, 33, 36syl2anc 643 1  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995    < clt 9120    <_ cle 9121    / cdiv 9677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678
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