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Theorem ledivge1le 9405
Description: If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)
Assertion
Ref Expression
ledivge1le  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( A  <_  B  ->  ( A  /  C
)  <_  B )
)

Proof of Theorem ledivge1le
StepHypRef Expression
1 divle1le 9404 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  <_  1  <->  A  <_  B ) )
21adantr 272 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  <->  A  <_  B ) )
3 rerpdivcl 9366 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
43adantr 272 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( A  /  B
)  e.  RR )
5 1red 7698 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  1  e.  RR )
6 rpre 9342 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  C  e.  RR )
76adantl 273 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  C  e.  RR )
8 letr 7763 . . . . . . . . . 10  |-  ( ( ( A  /  B
)  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( ( A  /  B )  <_  1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
94, 5, 7, 8syl3anc 1197 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( ( A  /  B )  <_ 
1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
109expd 256 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  ->  ( 1  <_  C  ->  ( A  /  B
)  <_  C )
) )
112, 10sylbird 169 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( A  <_  B  ->  ( 1  <_  C  ->  ( A  /  B
)  <_  C )
) )
1211com23 78 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( 1  <_  C  ->  ( A  <_  B  ->  ( A  /  B
)  <_  C )
) )
1312expimpd 358 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( C  e.  RR+  /\  1  <_  C
)  ->  ( A  <_  B  ->  ( A  /  B )  <_  C
) ) )
1413ex 114 . . . 4  |-  ( A  e.  RR  ->  ( B  e.  RR+  ->  (
( C  e.  RR+  /\  1  <_  C )  ->  ( A  <_  B  ->  ( A  /  B
)  <_  C )
) ) )
15143imp1 1179 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  B
)  <_  C )
16 simp1 962 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  ->  A  e.  RR )
176adantr 272 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  C  e.  RR )
18 0lt1 7805 . . . . . . . . . 10  |-  0  <  1
19 0red 7684 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  0  e.  RR )
20 1red 7698 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  1  e.  RR )
21 ltletr 7769 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( 0  <  1  /\  1  <_  C )  ->  0  <  C
) )
2219, 20, 6, 21syl3anc 1197 . . . . . . . . . 10  |-  ( C  e.  RR+  ->  ( ( 0  <  1  /\  1  <_  C )  ->  0  <  C ) )
2318, 22mpani 424 . . . . . . . . 9  |-  ( C  e.  RR+  ->  ( 1  <_  C  ->  0  <  C ) )
2423imp 123 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  0  <  C )
2517, 24jca 302 . . . . . . 7  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  ( C  e.  RR  /\  0  <  C ) )
26253ad2ant3 985 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( C  e.  RR  /\  0  <  C ) )
27 rpregt0 9349 . . . . . . 7  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
28273ad2ant2 984 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( B  e.  RR  /\  0  <  B ) )
2916, 26, 283jca 1142 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C )  /\  ( B  e.  RR  /\  0  < 
B ) ) )
3029adantr 272 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C )  /\  ( B  e.  RR  /\  0  < 
B ) ) )
31 lediv23 8554 . . . 4  |-  ( ( A  e.  RR  /\  ( C  e.  RR  /\  0  <  C )  /\  ( B  e.  RR  /\  0  < 
B ) )  -> 
( ( A  /  C )  <_  B  <->  ( A  /  B )  <_  C ) )
3230, 31syl 14 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( ( A  /  C )  <_  B  <->  ( A  /  B )  <_  C ) )
3315, 32mpbird 166 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  C
)  <_  B )
3433ex 114 1  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( A  <_  B  ->  ( A  /  C
)  <_  B )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 943    e. wcel 1461   class class class wbr 3893  (class class class)co 5726   RRcr 7539   0cc0 7540   1c1 7541    < clt 7717    <_ cle 7718    / cdiv 8338   RR+crp 9336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7629  ax-resscn 7630  ax-1cn 7631  ax-1re 7632  ax-icn 7633  ax-addcl 7634  ax-addrcl 7635  ax-mulcl 7636  ax-mulrcl 7637  ax-addcom 7638  ax-mulcom 7639  ax-addass 7640  ax-mulass 7641  ax-distr 7642  ax-i2m1 7643  ax-0lt1 7644  ax-1rid 7645  ax-0id 7646  ax-rnegex 7647  ax-precex 7648  ax-cnre 7649  ax-pre-ltirr 7650  ax-pre-ltwlin 7651  ax-pre-lttrn 7652  ax-pre-apti 7653  ax-pre-ltadd 7654  ax-pre-mulgt0 7655  ax-pre-mulext 7656
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7719  df-mnf 7720  df-xr 7721  df-ltxr 7722  df-le 7723  df-sub 7851  df-neg 7852  df-reap 8248  df-ap 8255  df-div 8339  df-rp 9337
This theorem is referenced by: (None)
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