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Theorem ledivp1i 10521
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 9631 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9645 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10499 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9746 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10449 . . . . . 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 9632 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9750 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 675 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 10138 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10362 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 17 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 lemul1 10446 . . . . . . . . . . 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 1347 . . . . . . . . . 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 435 . . . . . . . . 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 680 . . . . . . . 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 37 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <_  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2423biimpd 210 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2514, 24syl 17 . . . . 5  |-  ( 0  <_  C  ->  ( A  <_  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2625imp 430 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) )
2716recni 9644 . . . . . . 7  |-  B  e.  CC
284recni 9644 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2927, 28divcan1zi 10332 . . . . . 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 466 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3226, 31breqtrd 4441 . . 3  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  B )
33323adant1 1023 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <_  B )
341, 2remulcli 9646 . . 3  |-  ( A  x.  C )  e.  RR
351, 4remulcli 9646 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3634, 35, 16letri 9752 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <_  B )  -> 
( A  x.  C
)  <_  B )
3711, 33, 36syl2anc 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 1867    =/= wne 2616   class class class wbr 4417  (class class class)co 6296   RRcr 9527   0cc0 9528   1c1 9529    + caddc 9531    x. cmul 9533    < clt 9664    <_ cle 9665    / cdiv 10258
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 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
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 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-div 10259
This theorem is referenced by: (None)
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