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Theorem lediv2a 9215
Description: Division of both sides of 'less than or equal to' into a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.)
Assertion
Ref Expression
lediv2a |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( C / B ) <_ ( C / A ) )
Proof of Theorem lediv2a
StepHypRef Expression
1 pm3.2 139 . . . . . . 7 |- ( C e. RR -> ( C e. RR -> ( C e. RR /\ C e. RR ) ) )
21pm2.43i 49 . . . . . 6 |- ( C e. RR -> ( C e. RR /\ C e. RR ) )
32adantr 276 . . . . 5 |- ( ( C e. RR /\ 0 <_ C ) -> ( C e. RR /\ C e. RR ) )
4 leid 8399 . . . . . . 7 |- ( C e. RR -> C <_ C )
54anim2i 342 . . . . . 6 |- ( ( 0 <_ C /\ C e. RR ) -> ( 0 <_ C /\ C <_ C ) )
65ancoms 268 . . . . 5 |- ( ( C e. RR /\ 0 <_ C ) -> ( 0 <_ C /\ C <_ C ) )
73, 6jca 306 . . . 4 |- ( ( C e. RR /\ 0 <_ C ) -> ( ( C e. RR /\ C e. RR ) /\ ( 0 <_ C /\ C <_ C ) ) )
87ad2antlr 493 . . 3 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( ( C e. RR /\ C e. RR ) /\ ( 0 <_ C /\ C <_ C ) ) )
983adantl2 1185 . 2 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( ( C e. RR /\ C e. RR ) /\ ( 0 <_ C /\ C <_ C ) ) )
10 id 19 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A e. RR /\ B e. RR ) )
1110ad2ant2r 513 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A e. RR /\ B e. RR ) )
1211adantr 276 . . . 4 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ A <_ B ) -> ( A e. RR /\ B e. RR ) )
13 simplr 533 . . . . 5 |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < A )
1413anim1i 340 . . . 4 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ A <_ B ) -> ( 0 < A /\ A <_ B ) )
1512, 14jca 306 . . 3 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ A <_ B ) -> ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ A <_ B ) ) )
16153adantl3 1186 . 2 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ A <_ B ) ) )
17 lediv12a 9214 . 2 |- ( ( ( ( C e. RR /\ C e. RR ) /\ ( 0 <_ C /\ C <_ C ) ) /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ A <_ B ) ) ) -> ( C / B ) <_ ( C / A ) )
189, 16, 17syl2anc 415 1 |- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 <_ C ) ) /\ A <_ B ) -> ( C / B ) <_ ( C / A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   < clt 8350   <_ cle 8351   / cdiv 8992
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993
This theorem is referenced by:  lediv2ad  10099
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