ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ledivge1le Unicode version

Theorem ledivge1le 9135
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 9134 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  <_  1  <->  A  <_  B ) )
21adantr 270 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  <->  A  <_  B ) )
3 rerpdivcl 9096 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
43adantr 270 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( A  /  B
)  e.  RR )
5 1red 7447 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  1  e.  RR )
6 rpre 9072 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  C  e.  RR )
76adantl 271 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  C  e.  RR )
8 letr 7512 . . . . . . . . . 10  |-  ( ( ( A  /  B
)  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( ( A  /  B )  <_  1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
94, 5, 7, 8syl3anc 1172 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( ( A  /  B )  <_ 
1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
109expd 254 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  ->  ( 1  <_  C  ->  ( A  /  B
)  <_  C )
) )
112, 10sylbird 168 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( A  <_  B  ->  ( 1  <_  C  ->  ( A  /  B
)  <_  C )
) )
1211com23 77 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( 1  <_  C  ->  ( A  <_  B  ->  ( A  /  B
)  <_  C )
) )
1312expimpd 355 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( C  e.  RR+  /\  1  <_  C
)  ->  ( A  <_  B  ->  ( A  /  B )  <_  C
) ) )
1413ex 113 . . . 4  |-  ( A  e.  RR  ->  ( B  e.  RR+  ->  (
( C  e.  RR+  /\  1  <_  C )  ->  ( A  <_  B  ->  ( A  /  B
)  <_  C )
) ) )
15143imp1 1154 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  B
)  <_  C )
16 simp1 941 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  ->  A  e.  RR )
176adantr 270 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  C  e.  RR )
18 0lt1 7554 . . . . . . . . . 10  |-  0  <  1
19 0red 7433 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  0  e.  RR )
20 1red 7447 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  1  e.  RR )
21 ltletr 7518 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( 0  <  1  /\  1  <_  C )  ->  0  <  C
) )
2219, 20, 6, 21syl3anc 1172 . . . . . . . . . 10  |-  ( C  e.  RR+  ->  ( ( 0  <  1  /\  1  <_  C )  ->  0  <  C ) )
2318, 22mpani 421 . . . . . . . . 9  |-  ( C  e.  RR+  ->  ( 1  <_  C  ->  0  <  C ) )
2423imp 122 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  0  <  C )
2517, 24jca 300 . . . . . . 7  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  ( C  e.  RR  /\  0  <  C ) )
26253ad2ant3 964 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( C  e.  RR  /\  0  <  C ) )
27 rpregt0 9079 . . . . . . 7  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
28273ad2ant2 963 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( B  e.  RR  /\  0  <  B ) )
2916, 26, 283jca 1121 . . . . 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 270 . . . 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 8289 . . . 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 165 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  C
)  <_  B )
3433ex 113 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 102    <-> wb 103    /\ w3a 922    e. wcel 1436   class class class wbr 3820  (class class class)co 5613   RRcr 7293   0cc0 7294   1c1 7295    < clt 7466    <_ cle 7467    / cdiv 8078   RR+crp 9066
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234  ax-setind 4326  ax-cnex 7380  ax-resscn 7381  ax-1cn 7382  ax-1re 7383  ax-icn 7384  ax-addcl 7385  ax-addrcl 7386  ax-mulcl 7387  ax-mulrcl 7388  ax-addcom 7389  ax-mulcom 7390  ax-addass 7391  ax-mulass 7392  ax-distr 7393  ax-i2m1 7394  ax-0lt1 7395  ax-1rid 7396  ax-0id 7397  ax-rnegex 7398  ax-precex 7399  ax-cnre 7400  ax-pre-ltirr 7401  ax-pre-ltwlin 7402  ax-pre-lttrn 7403  ax-pre-apti 7404  ax-pre-ltadd 7405  ax-pre-mulgt0 7406  ax-pre-mulext 7407
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-po 4097  df-iso 4098  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-iota 4946  df-fun 4983  df-fv 4989  df-riota 5569  df-ov 5616  df-oprab 5617  df-mpt2 5618  df-pnf 7468  df-mnf 7469  df-xr 7470  df-ltxr 7471  df-le 7472  df-sub 7599  df-neg 7600  df-reap 7993  df-ap 8000  df-div 8079  df-rp 9067
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator