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Theorem lt2addi 7502
Description: Adding both side of two inequalities. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 14-May-1999.)
Hypotheses
Ref Expression
lt2.1  |-  A  e.  RR
lt2.2  |-  B  e.  RR
lt2.3  |-  C  e.  RR
lt.4  |-  D  e.  RR
Assertion
Ref Expression
lt2addi  |-  ( ( A  <  C  /\  B  <  D )  -> 
( A  +  B
)  <  ( C  +  D ) )

Proof of Theorem lt2addi
StepHypRef Expression
1 lt2.1 . 2  |-  A  e.  RR
2 lt2.2 . 2  |-  B  e.  RR
3 lt2.3 . 2  |-  C  e.  RR
4 lt.4 . 2  |-  D  e.  RR
5 lt2add 7440 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  < 
C  /\  B  <  D )  ->  ( A  +  B )  <  ( C  +  D )
) )
61, 2, 3, 4, 5mp4an 403 1  |-  ( ( A  <  C  /\  B  <  D )  -> 
( A  +  B
)  <  ( C  +  D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888    + caddc 6892    < clt 7060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-iota 4867  df-fv 4910  df-ov 5515  df-pnf 7062  df-mnf 7063  df-ltxr 7065
This theorem is referenced by: (None)
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