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Theorem ledivp1i 10279
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 9406 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9420 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10257 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9518 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10205 . . . . . 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 1304 . . . 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 1009 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  C )  <_  ( A  x.  ( C  +  1 ) ) )
12 0re 9407 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9522 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 671 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9896 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10118 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 16 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 lemul1 10202 . . . . . . . . . . 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 1301 . . . . . . . . . 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 9419 . . . . . . 7  |-  B  e.  CC
284recni 9419 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2927, 28divcan1zi 10088 . . . . . 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 4337 . . 3  |-  ( ( 0  <_  C  /\  A  <_  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <_  B )
33323adant1 1006 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <_  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <_  B )
341, 2remulcli 9421 . . 3  |-  ( A  x.  C )  e.  RR
351, 4remulcli 9421 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3634, 35, 16letri 9524 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2620   class class class wbr 4313  (class class class)co 6112   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    < clt 9439    <_ cle 9440    / cdiv 10014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015
This theorem is referenced by: (None)
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