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Theorem ltdivp1i 10373
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 9499 . . . . . 6  |-  1  e.  RR
42, 3readdcli 9513 . . . . 5  |-  ( C  +  1 )  e.  RR
52ltp1i 10350 . . . . . . 7  |-  C  < 
( C  +  1 )
62, 4, 5ltleii 9611 . . . . . 6  |-  C  <_ 
( C  +  1 )
7 lemul2a 10298 . . . . . 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 1305 . . . 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 9500 . . . . . . . 8  |-  0  e.  RR
1312, 2, 4lelttri 9615 . . . . . . 7  |-  ( ( 0  <_  C  /\  C  <  ( C  + 
1 ) )  -> 
0  <  ( C  +  1 ) )
145, 13mpan2 671 . . . . . 6  |-  ( 0  <_  C  ->  0  <  ( C  +  1 ) )
154gt0ne0i 9989 . . . . . . . . 9  |-  ( 0  <  ( C  + 
1 )  ->  ( C  +  1 )  =/=  0 )
16 prodgt0.2 . . . . . . . . . 10  |-  B  e.  RR
1716, 4redivclzi 10211 . . . . . . . . 9  |-  ( ( C  +  1 )  =/=  0  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
1815, 17syl 16 . . . . . . . 8  |-  ( 0  <  ( C  + 
1 )  ->  ( B  /  ( C  + 
1 ) )  e.  RR )
19 ltmul1 10293 . . . . . . . . . 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 1302 . . . . . . . . 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 683 . . . . . . . 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 669 . . . . . . 7  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  <->  ( A  x.  ( C  +  1 ) )  <  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) ) ) )
2322biimpd 207 . . . . . 6  |-  ( 0  <  ( C  + 
1 )  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2414, 23syl 16 . . . . 5  |-  ( 0  <_  C  ->  ( A  <  ( B  / 
( C  +  1 ) )  ->  ( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  + 
1 ) )  x.  ( C  +  1 ) ) ) )
2524imp 429 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  ( ( B  /  ( C  +  1 ) )  x.  ( C  + 
1 ) ) )
2616recni 9512 . . . . . . 7  |-  B  e.  CC
274recni 9512 . . . . . . 7  |-  ( C  +  1 )  e.  CC
2826, 27divcan1zi 10181 . . . . . 6  |-  ( ( C  +  1 )  =/=  0  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
2914, 15, 283syl 20 . . . . 5  |-  ( 0  <_  C  ->  (
( B  /  ( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3029adantr 465 . . . 4  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( ( B  / 
( C  +  1 ) )  x.  ( C  +  1 ) )  =  B )
3125, 30breqtrd 4427 . . 3  |-  ( ( 0  <_  C  /\  A  <  ( B  / 
( C  +  1 ) ) )  -> 
( A  x.  ( C  +  1 ) )  <  B )
32313adant1 1006 . 2  |-  ( ( 0  <_  A  /\  0  <_  C  /\  A  <  ( B  /  ( C  +  1 ) ) )  ->  ( A  x.  ( C  +  1 ) )  <  B )
331, 2remulcli 9514 . . 3  |-  ( A  x.  C )  e.  RR
341, 4remulcli 9514 . . 3  |-  ( A  x.  ( C  + 
1 ) )  e.  RR
3533, 34, 16lelttri 9615 . 2  |-  ( ( ( A  x.  C
)  <_  ( A  x.  ( C  +  1 ) )  /\  ( A  x.  ( C  +  1 ) )  <  B )  -> 
( A  x.  C
)  <  B )
3611, 32, 35syl2anc 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 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403  (class class class)co 6203   RRcr 9395   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401    < clt 9532    <_ cle 9533    / cdiv 10107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-po 4752  df-so 4753  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108
This theorem is referenced by: (None)
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