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Theorem le2add 9502
Description: Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
le2add  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <_  D
)  ->  ( A  +  B )  <_  ( C  +  D )
) )

Proof of Theorem le2add
StepHypRef Expression
1 simpll 731 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
2 simprl 733 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR )
3 simplr 732 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR )
4 leadd1 9488 . . . 4  |-  ( ( A  e.  RR  /\  C  e.  RR  /\  B  e.  RR )  ->  ( A  <_  C  <->  ( A  +  B )  <_  ( C  +  B )
) )
51, 2, 3, 4syl3anc 1184 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  <_  C  <->  ( A  +  B )  <_  ( C  +  B ) ) )
6 simprr 734 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
7 leadd2 9489 . . . 4  |-  ( ( B  e.  RR  /\  D  e.  RR  /\  C  e.  RR )  ->  ( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D )
) )
83, 6, 2, 7syl3anc 1184 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( B  <_  D  <->  ( C  +  B )  <_  ( C  +  D ) ) )
95, 8anbi12d 692 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <_  D
)  <->  ( ( A  +  B )  <_ 
( C  +  B
)  /\  ( C  +  B )  <_  ( C  +  D )
) ) )
101, 3readdcld 9107 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( A  +  B
)  e.  RR )
112, 3readdcld 9107 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  B
)  e.  RR )
122, 6readdcld 9107 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( C  +  D
)  e.  RR )
13 letr 9159 . . 3  |-  ( ( ( A  +  B
)  e.  RR  /\  ( C  +  B
)  e.  RR  /\  ( C  +  D
)  e.  RR )  ->  ( ( ( A  +  B )  <_  ( C  +  B )  /\  ( C  +  B )  <_  ( C  +  D
) )  ->  ( A  +  B )  <_  ( C  +  D
) ) )
1410, 11, 12, 13syl3anc 1184 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( ( A  +  B )  <_ 
( C  +  B
)  /\  ( C  +  B )  <_  ( C  +  D )
)  ->  ( A  +  B )  <_  ( C  +  D )
) )
159, 14sylbid 207 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <_  D
)  ->  ( A  +  B )  <_  ( C  +  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   class class class wbr 4204  (class class class)co 6073   RRcr 8981    + caddc 8985    <_ cle 9113
This theorem is referenced by:  addge0  9509  le2addi  9582  le2addd  9636  sqrlem7  12046  lo1add  12412  climcndslem1  12621  climcndslem2  12622  mdegmullem  19993  mumullem2  20955  pntrsumbnd2  21253  pntlemf  21291  ubthlem2  22365  nmoptrii  23589  cdj3i  23936  itg2addnc  26249  fzadd2  26426  jm2.26lem3  27053  swrdccatin2  28166  2cshwmod  28213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118
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