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Theorem ledivp1i 10472
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 9596 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9610 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10450 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9708 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10398 . . . . . 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 671 . . . . 5  |-  ( ( C  e.  RR  /\  ( C  +  1
)  e.  RR  /\  ( A  e.  RR  /\  0  <_  A )
)  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
92, 4, 8mp3an12 1314 . . . 4  |-  ( ( A  e.  RR  /\  0  <_  A )  -> 
( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) ) )
101, 9mpan 670 . . 3  |-  ( 0  <_  A  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
11103ad2ant1 1017 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 9597 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9712 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 671 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 10089 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10311 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 16 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 lemul1 10395 . . . . . . . . . . 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 1311 . . . . . . . . . 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 434 . . . . . . . . 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 676 . . . . . . . 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 36 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <_  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <_  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2423biimpd 207 . . . . . 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 429 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) )
2716recni 9609 . . . . . . 7  |-  B  e.  CC
284recni 9609 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2927, 28divcan1zi 10281 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3014, 15, 293syl 20 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3130adantr 465 . . . 4  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3226, 31breqtrd 4471 . . 3  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  B )
33323adant1 1014 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <_  B )
341, 2remulcli 9611 . . 3  |-  ( A  x.  C )  e.  RR
351, 4remulcli 9611 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3634, 35, 16letri 9714 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <_  B )  -> 
( A  x.  C
)  <_  B )
3711, 33, 36syl2anc 661 1  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6285   RRcr 9492   0cc0 9493   1c1 9494    + caddc 9496    x. cmul 9498    < clt 9629    <_ cle 9630    / cdiv 10207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-div 10208
This theorem is referenced by: (None)
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