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Theorem addextpr 7452
Description: Strong extensionality of addition (ordering version). This is similar to addext 8395 but for positive reals and based on less-than rather than apartness. (Contributed by Jim Kingdon, 17-Feb-2020.)
Assertion
Ref Expression
addextpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( ( A  +P.  B )  <P 
( C  +P.  D
)  ->  ( A  <P  C  \/  B  <P  D ) ) )

Proof of Theorem addextpr
Dummy variables  f  g are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addclpr 7368 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21adantr 274 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  e.  P. )
3 addclpr 7368 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( C  +P.  D
)  e.  P. )
43adantl 275 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  D )  e.  P. )
5 simprl 521 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  C  e.  P. )
6 simplr 520 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  B  e.  P. )
7 addclpr 7368 . . . 4  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
85, 6, 7syl2anc 409 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  e.  P. )
9 ltsopr 7427 . . . 4  |-  <P  Or  P.
10 sowlin 4249 . . . 4  |-  ( ( 
<P  Or  P.  /\  (
( A  +P.  B
)  e.  P.  /\  ( C  +P.  D )  e.  P.  /\  ( C  +P.  B )  e. 
P. ) )  -> 
( ( A  +P.  B )  <P  ( C  +P.  D )  ->  (
( A  +P.  B
)  <P  ( C  +P.  B )  \/  ( C  +P.  B )  <P 
( C  +P.  D
) ) ) )
119, 10mpan 421 . . 3  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( C  +P.  D )  e.  P.  /\  ( C  +P.  B )  e. 
P. )  ->  (
( A  +P.  B
)  <P  ( C  +P.  D )  ->  ( ( A  +P.  B )  <P 
( C  +P.  B
)  \/  ( C  +P.  B )  <P 
( C  +P.  D
) ) ) )
122, 4, 8, 11syl3anc 1217 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( ( A  +P.  B )  <P 
( C  +P.  D
)  ->  ( ( A  +P.  B )  <P 
( C  +P.  B
)  \/  ( C  +P.  B )  <P 
( C  +P.  D
) ) ) )
13 simpll 519 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  A  e.  P. )
14 ltaprg 7450 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  ( A  <P  C  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
1513, 5, 6, 14syl3anc 1217 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  <P  C  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
16 addcomprg 7409 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1716adantl 275 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  /\  ( f  e.  P.  /\  g  e.  P. )
)  ->  ( f  +P.  g )  =  ( g  +P.  f ) )
1817, 13, 6caovcomd 5934 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  =  ( B  +P.  A ) )
1917, 5, 6caovcomd 5934 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  =  ( B  +P.  C ) )
2018, 19breq12d 3949 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( ( A  +P.  B )  <P 
( C  +P.  B
)  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
2115, 20bitr4d 190 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  <P  C  <->  ( A  +P.  B )  <P  ( C  +P.  B ) ) )
22 simprr 522 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  D  e.  P. )
23 ltaprg 7450 . . . 4  |-  ( ( B  e.  P.  /\  D  e.  P.  /\  C  e.  P. )  ->  ( B  <P  D  <->  ( C  +P.  B )  <P  ( C  +P.  D ) ) )
246, 22, 5, 23syl3anc 1217 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( B  <P  D  <->  ( C  +P.  B )  <P  ( C  +P.  D ) ) )
2521, 24orbi12d 783 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( ( A  <P  C  \/  B  <P  D )  <->  ( ( A  +P.  B )  <P 
( C  +P.  B
)  \/  ( C  +P.  B )  <P 
( C  +P.  D
) ) ) )
2612, 25sylibrd 168 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( ( A  +P.  B )  <P 
( C  +P.  D
)  ->  ( A  <P  C  \/  B  <P  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 698    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3936    Or wor 4224  (class class class)co 5781   P.cnp 7122    +P. cpp 7124    <P cltp 7126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-iplp 7299  df-iltp 7301
This theorem is referenced by:  mulextsr1lem  7611
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