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Theorem addextpr 7371
Description: Strong extensionality of addition (ordering version). This is similar to addext 8284 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 7287 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21adantr 272 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  e.  P. )
3 addclpr 7287 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( C  +P.  D
)  e.  P. )
43adantl 273 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  D )  e.  P. )
5 simprl 503 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  C  e.  P. )
6 simplr 502 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  B  e.  P. )
7 addclpr 7287 . . . 4  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
85, 6, 7syl2anc 406 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  e.  P. )
9 ltsopr 7346 . . . 4  |-  <P  Or  P.
10 sowlin 4200 . . . 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 418 . . 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 1197 . 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 501 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  A  e.  P. )
14 ltaprg 7369 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  ( A  <P  C  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
1513, 5, 6, 14syl3anc 1197 . . . 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 7328 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1716adantl 273 . . . . . 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 5879 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  =  ( B  +P.  A ) )
1917, 5, 6caovcomd 5879 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  =  ( B  +P.  C ) )
2018, 19breq12d 3906 . . . 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 504 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  D  e.  P. )
23 ltaprg 7369 . . . 4  |-  ( ( B  e.  P.  /\  D  e.  P.  /\  C  e.  P. )  ->  ( B  <P  D  <->  ( C  +P.  B )  <P  ( C  +P.  D ) ) )
246, 22, 5, 23syl3anc 1197 . . 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 765 . 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 680    /\ w3a 943    = wceq 1312    e. wcel 1461   class class class wbr 3893    Or wor 4175  (class class class)co 5726   P.cnp 7041    +P. cpp 7043    <P cltp 7045
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-iinf 4460
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-eprel 4169  df-id 4173  df-po 4176  df-iso 4177  df-iord 4246  df-on 4248  df-suc 4251  df-iom 4463  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-ov 5729  df-oprab 5730  df-mpo 5731  df-1st 5990  df-2nd 5991  df-recs 6154  df-irdg 6219  df-1o 6265  df-2o 6266  df-oadd 6269  df-omul 6270  df-er 6381  df-ec 6383  df-qs 6387  df-ni 7054  df-pli 7055  df-mi 7056  df-lti 7057  df-plpq 7094  df-mpq 7095  df-enq 7097  df-nqqs 7098  df-plqqs 7099  df-mqqs 7100  df-1nqqs 7101  df-rq 7102  df-ltnqqs 7103  df-enq0 7174  df-nq0 7175  df-0nq0 7176  df-plq0 7177  df-mq0 7178  df-inp 7216  df-iplp 7218  df-iltp 7220
This theorem is referenced by:  mulextsr1lem  7516
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