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Theorem ltaddpr2 8912
Description: The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltaddpr2  |-  ( C  e.  P.  ->  (
( A  +P.  B
)  =  C  ->  A  <P  C ) )

Proof of Theorem ltaddpr2
StepHypRef Expression
1 eleq1 2496 . . . 4  |-  ( ( A  +P.  B )  =  C  ->  (
( A  +P.  B
)  e.  P.  <->  C  e.  P. ) )
2 dmplp 8889 . . . . 5  |-  dom  +P.  =  ( P.  X.  P. )
3 0npr 8869 . . . . 5  |-  -.  (/)  e.  P.
42, 3ndmovrcl 6233 . . . 4  |-  ( ( A  +P.  B )  e.  P.  ->  ( A  e.  P.  /\  B  e.  P. ) )
51, 4syl6bir 221 . . 3  |-  ( ( A  +P.  B )  =  C  ->  ( C  e.  P.  ->  ( A  e.  P.  /\  B  e.  P. )
) )
6 ltaddpr 8911 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
7 breq2 4216 . . . 4  |-  ( ( A  +P.  B )  =  C  ->  ( A  <P  ( A  +P.  B )  <->  A  <P  C ) )
86, 7syl5ib 211 . . 3  |-  ( ( A  +P.  B )  =  C  ->  (
( A  e.  P.  /\  B  e.  P. )  ->  A  <P  C )
)
95, 8syld 42 . 2  |-  ( ( A  +P.  B )  =  C  ->  ( C  e.  P.  ->  A 
<P  C ) )
109com12 29 1  |-  ( C  e.  P.  ->  (
( A  +P.  B
)  =  C  ->  A  <P  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   P.cnp 8734    +P. cpp 8736    <P cltp 8738
This theorem is referenced by:  mulgt0sr  8980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-omul 6729  df-er 6905  df-ni 8749  df-pli 8750  df-mi 8751  df-lti 8752  df-plpq 8785  df-mpq 8786  df-ltpq 8787  df-enq 8788  df-nq 8789  df-erq 8790  df-plq 8791  df-mq 8792  df-1nq 8793  df-rq 8794  df-ltnq 8795  df-np 8858  df-plp 8860  df-ltp 8862
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