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Theorem ledivge1le 9739
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 9738 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( A  /  B )  <_  1  <->  A  <_  B ) )
21adantr 276 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  <->  A  <_  B ) )
3 rerpdivcl 9697 . . . . . . . . . . 11  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( A  /  B
)  e.  RR )
43adantr 276 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( A  /  B
)  e.  RR )
5 1red 7985 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  1  e.  RR )
6 rpre 9673 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  C  e.  RR )
76adantl 277 . . . . . . . . . 10  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  C  e.  RR )
8 letr 8053 . . . . . . . . . 10  |-  ( ( ( A  /  B
)  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( ( A  /  B )  <_  1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
94, 5, 7, 8syl3anc 1248 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( ( A  /  B )  <_ 
1  /\  1  <_  C )  ->  ( A  /  B )  <_  C
) )
109expd 258 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR+ )  /\  C  e.  RR+ )  ->  ( ( A  /  B )  <_  1  ->  ( 1  <_  C  ->  ( A  /  B
)  <_  C )
) )
112, 10sylbird 170 . . . . . . 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 363 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR+ )  -> 
( ( C  e.  RR+  /\  1  <_  C
)  ->  ( A  <_  B  ->  ( A  /  B )  <_  C
) ) )
1413ex 115 . . . 4  |-  ( A  e.  RR  ->  ( B  e.  RR+  ->  (
( C  e.  RR+  /\  1  <_  C )  ->  ( A  <_  B  ->  ( A  /  B
)  <_  C )
) ) )
15143imp1 1221 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  B
)  <_  C )
16 simp1 998 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  ->  A  e.  RR )
176adantr 276 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  C  e.  RR )
18 0lt1 8097 . . . . . . . . . 10  |-  0  <  1
19 0red 7971 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  0  e.  RR )
20 1red 7985 . . . . . . . . . . 11  |-  ( C  e.  RR+  ->  1  e.  RR )
21 ltletr 8060 . . . . . . . . . . 11  |-  ( ( 0  e.  RR  /\  1  e.  RR  /\  C  e.  RR )  ->  (
( 0  <  1  /\  1  <_  C )  ->  0  <  C
) )
2219, 20, 6, 21syl3anc 1248 . . . . . . . . . 10  |-  ( C  e.  RR+  ->  ( ( 0  <  1  /\  1  <_  C )  ->  0  <  C ) )
2318, 22mpani 430 . . . . . . . . 9  |-  ( C  e.  RR+  ->  ( 1  <_  C  ->  0  <  C ) )
2423imp 124 . . . . . . . 8  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  0  <  C )
2517, 24jca 306 . . . . . . 7  |-  ( ( C  e.  RR+  /\  1  <_  C )  ->  ( C  e.  RR  /\  0  <  C ) )
26253ad2ant3 1021 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( C  e.  RR  /\  0  <  C ) )
27 rpregt0 9680 . . . . . . 7  |-  ( B  e.  RR+  ->  ( B  e.  RR  /\  0  <  B ) )
28273ad2ant2 1020 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  -> 
( B  e.  RR  /\  0  <  B ) )
2916, 26, 283jca 1178 . . . . 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 276 . . . 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 8863 . . . 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 167 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR+  /\  ( C  e.  RR+  /\  1  <_  C ) )  /\  A  <_  B )  -> 
( A  /  C
)  <_  B )
3433ex 115 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 104    <-> wb 105    /\ w3a 979    e. wcel 2158   class class class wbr 4015  (class class class)co 5888   RRcr 7823   0cc0 7824   1c1 7825    < clt 8005    <_ cle 8006    / cdiv 8642   RR+crp 9666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-po 4308  df-iso 4309  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-rp 9667
This theorem is referenced by: (None)
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