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Theorem ltdivp1i 9699
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 8853 . . . . . 6  |-  1  e.  RR
42, 3readdcli 8866 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 9676 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 8957 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 9627 . . . . . 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 8854 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 8962 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 652 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9324 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 9542 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 15 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 ltmul1 9622 . . . . . . . . . 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 1264 . . . . . . . . 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 664 . . . . . . . 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 650 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2322biimpd 198 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2414, 23syl 15 . . . . 5  |-  ( 0  <_  C  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2524imp 418 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  +  1 ) )  x.  ( C  + 
1 ) ) )
2616recni 8865 . . . . . . 7  |-  B  e.  CC
274recni 8865 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2826, 27divcan1zi 9512 . . . . . 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 451 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3125, 30breqtrd 4063 . . 3  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  B )
32313adant1 973 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <  B )
331, 2remulcli 8867 . . 3  |-  ( A  x.  C )  e.  RR
341, 4remulcli 8867 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3533, 34, 16lelttri 8962 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <  B )  -> 
( A  x.  C
)  <  B )
3611, 32, 35syl2anc 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 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884    / cdiv 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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