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Theorem addextpr 7633
Description: Strong extensionality of addition (ordering version). This is similar to addext 8580 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 7549 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
21adantr 276 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  e.  P. )
3 addclpr 7549 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( C  +P.  D
)  e.  P. )
43adantl 277 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  D )  e.  P. )
5 simprl 529 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  C  e.  P. )
6 simplr 528 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  B  e.  P. )
7 addclpr 7549 . . . 4  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( C  +P.  B
)  e.  P. )
85, 6, 7syl2anc 411 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  e.  P. )
9 ltsopr 7608 . . . 4  |-  <P  Or  P.
10 sowlin 4332 . . . 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 424 . . 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 1248 . 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 527 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  A  e.  P. )
14 ltaprg 7631 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  ( A  <P  C  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
1513, 5, 6, 14syl3anc 1248 . . . 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 7590 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
1716adantl 277 . . . . . 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 6044 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  =  ( B  +P.  A ) )
1917, 5, 6caovcomd 6044 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  =  ( B  +P.  C ) )
2018, 19breq12d 4028 . . . 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 191 . . 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 531 . . . 4  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  D  e.  P. )
23 ltaprg 7631 . . . 4  |-  ( ( B  e.  P.  /\  D  e.  P.  /\  C  e.  P. )  ->  ( B  <P  D  <->  ( C  +P.  B )  <P  ( C  +P.  D ) ) )
246, 22, 5, 23syl3anc 1248 . . 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 794 . 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 169 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 104    <-> wb 105    \/ wo 709    /\ w3a 979    = wceq 1363    e. wcel 2158   class class class wbr 4015    Or wor 4307  (class class class)co 5888   P.cnp 7303    +P. cpp 7305    <P cltp 7307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-2o 6431  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-pli 7317  df-mi 7318  df-lti 7319  df-plpq 7356  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-plqqs 7361  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-enq0 7436  df-nq0 7437  df-0nq0 7438  df-plq0 7439  df-mq0 7440  df-inp 7478  df-iplp 7480  df-iltp 7482
This theorem is referenced by:  mulextsr1lem  7792
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