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Theorem leltadd 10094
Description: Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)
Assertion
Ref Expression
leltadd  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( A  +  B )  <  ( C  +  D )
) )

Proof of Theorem leltadd
StepHypRef Expression
1 ltleadd 10093 . . . . 5  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( B  < 
D  /\  A  <_  C )  ->  ( B  +  A )  <  ( D  +  C )
) )
21ancomsd 455 . . . 4  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( D  e.  RR  /\  C  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
32ancom2s 809 . . 3  |-  ( ( ( B  e.  RR  /\  A  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
43ancom1s 812 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  <_  C  /\  B  <  D
)  ->  ( B  +  A )  <  ( D  +  C )
) )
5 recn 9625 . . . 4  |-  ( A  e.  RR  ->  A  e.  CC )
6 recn 9625 . . . 4  |-  ( B  e.  RR  ->  B  e.  CC )
7 addcom 9815 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B
)  =  ( B  +  A ) )
85, 6, 7syl2an 479 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  =  ( B  +  A ) )
9 recn 9625 . . . 4  |-  ( C  e.  RR  ->  C  e.  CC )
10 recn 9625 . . . 4  |-  ( D  e.  RR  ->  D  e.  CC )
11 addcom 9815 . . . 4  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  +  D
)  =  ( D  +  C ) )
129, 10, 11syl2an 479 . . 3  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  +  D
)  =  ( D  +  C ) )
138, 12breqan12d 4433 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  RR  /\  D  e.  RR ) )  -> 
( ( A  +  B )  <  ( C  +  D )  <->  ( B  +  A )  <  ( D  +  C ) ) )
144, 13sylibrd 237 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   class class class wbr 4417  (class class class)co 6297   CCcc 9533   RRcr 9534    + caddc 9538    < clt 9671    <_ cle 9672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6589  ax-resscn 9592  ax-1cn 9593  ax-icn 9594  ax-addcl 9595  ax-addrcl 9596  ax-mulcl 9597  ax-mulrcl 9598  ax-mulcom 9599  ax-addass 9600  ax-mulass 9601  ax-distr 9602  ax-i2m1 9603  ax-1ne0 9604  ax-1rid 9605  ax-rnegex 9606  ax-rrecex 9607  ax-cnre 9608  ax-pre-lttri 9609  ax-pre-lttrn 9610  ax-pre-ltadd 9611
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-po 4767  df-so 4768  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-iota 5557  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6300  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9673  df-mnf 9674  df-xr 9675  df-ltxr 9676  df-le 9677
This theorem is referenced by:  lt2add  10095  addgegt0  10097  leltaddd  10231  fldiv  12080
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