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Theorem addextpr 7638
Description: Strong extensionality of addition (ordering version). This is similar to addext 8585 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 7554 . . . 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 7554 . . . 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 7554 . . . 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 7613 . . . 4  |-  <P  Or  P.
10 sowlin 4335 . . . 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 1249 . 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 7636 . . . . 5  |-  ( ( A  e.  P.  /\  C  e.  P.  /\  B  e.  P. )  ->  ( A  <P  C  <->  ( B  +P.  A )  <P  ( B  +P.  C ) ) )
1513, 5, 6, 14syl3anc 1249 . . . 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 7595 . . . . . . 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 6048 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( A  +P.  B )  =  ( B  +P.  A ) )
1917, 5, 6caovcomd 6048 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( C  +P.  B )  =  ( B  +P.  C ) )
2018, 19breq12d 4031 . . . 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 7636 . . . 4  |-  ( ( B  e.  P.  /\  D  e.  P.  /\  C  e.  P. )  ->  ( B  <P  D  <->  ( C  +P.  B )  <P  ( C  +P.  D ) ) )
246, 22, 5, 23syl3anc 1249 . . 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 980    = wceq 1364    e. wcel 2160   class class class wbr 4018    Or wor 4310  (class class class)co 5891   P.cnp 7308    +P. cpp 7310    <P cltp 7312
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4304  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-ov 5894  df-oprab 5895  df-mpo 5896  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-1o 6435  df-2o 6436  df-oadd 6439  df-omul 6440  df-er 6553  df-ec 6555  df-qs 6559  df-ni 7321  df-pli 7322  df-mi 7323  df-lti 7324  df-plpq 7361  df-mpq 7362  df-enq 7364  df-nqqs 7365  df-plqqs 7366  df-mqqs 7367  df-1nqqs 7368  df-rq 7369  df-ltnqqs 7370  df-enq0 7441  df-nq0 7442  df-0nq0 7443  df-plq0 7444  df-mq0 7445  df-inp 7483  df-iplp 7485  df-iltp 7487
This theorem is referenced by:  mulextsr1lem  7797
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