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Theorem ltaddpr 8911
Description: The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltaddpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )

Proof of Theorem ltaddpr
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prn0 8866 . . . . 5  |-  ( B  e.  P.  ->  B  =/=  (/) )
2 n0 3637 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2sylib 189 . . . 4  |-  ( B  e.  P.  ->  E. y 
y  e.  B )
43adantl 453 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. y  y  e.  B )
5 addclpr 8895 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
65adantr 452 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( A  +P.  B )  e. 
P. )
7 df-plp 8860 . . . . . . . . . . . . 13  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
8 addclnq 8822 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
97, 8genpprecl 8878 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) ) )
109imp 419 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) )
11 elprnq 8868 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  +Q  y )  e.  Q. )
12 addnqf 8825 . . . . . . . . . . . . . . 15  |-  +Q  :
( Q.  X.  Q. )
--> Q.
1312fdmi 5596 . . . . . . . . . . . . . 14  |-  dom  +Q  =  ( Q.  X.  Q. )
14 0nnq 8801 . . . . . . . . . . . . . 14  |-  -.  (/)  e.  Q.
1513, 14ndmovrcl 6233 . . . . . . . . . . . . 13  |-  ( ( x  +Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
16 ltaddnq 8851 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  <Q  ( x  +Q  y ) )
1711, 15, 163syl 19 . . . . . . . . . . . 12  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  x  <Q  (
x  +Q  y ) )
18 prcdnq 8870 . . . . . . . . . . . 12  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  <Q  ( x  +Q  y )  ->  x  e.  ( A  +P.  B ) ) )
1917, 18mpd 15 . . . . . . . . . . 11  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  x  e.  ( A  +P.  B ) )
206, 10, 19syl2anc 643 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  x  e.  ( A  +P.  B
) )
2120exp32 589 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  e.  ( A  +P.  B ) ) ) )
2221com23 74 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  x  e.  ( A  +P.  B ) ) ) )
2322alrimdv 1643 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) ) )
24 dfss2 3337 . . . . . . 7  |-  ( A 
C_  ( A  +P.  B )  <->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) )
2523, 24syl6ibr 219 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C_  ( A  +P.  B ) ) )
26 vex 2959 . . . . . . . . 9  |-  y  e. 
_V
2726prlem934 8910 . . . . . . . 8  |-  ( A  e.  P.  ->  E. x  e.  A  -.  (
x  +Q  y )  e.  A )
2827adantr 452 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. x  e.  A  -.  ( x  +Q  y
)  e.  A )
29 eleq2 2497 . . . . . . . . . . . . 13  |-  ( A  =  ( A  +P.  B )  ->  ( (
x  +Q  y )  e.  A  <->  ( x  +Q  y )  e.  ( A  +P.  B ) ) )
3029biimprcd 217 . . . . . . . . . . . 12  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( A  =  ( A  +P.  B )  ->  ( x  +Q  y )  e.  A
) )
3130con3d 127 . . . . . . . . . . 11  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) )
329, 31syl6 31 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) )
3332exp3a 426 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( -.  ( x  +Q  y )  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3433com34 79 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( -.  ( x  +Q  y )  e.  A  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3534rexlimdv 2829 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. x  e.  A  -.  ( x  +Q  y )  e.  A  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) ) )
3628, 35mpd 15 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) )
3725, 36jcad 520 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B ) ) ) )
38 dfpss2 3432 . . . . 5  |-  ( A 
C.  ( A  +P.  B )  <->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B
) ) )
3937, 38syl6ibr 219 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C.  ( A  +P.  B ) ) )
4039exlimdv 1646 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  y  e.  B  ->  A  C.  ( A  +P.  B
) ) )
414, 40mpd 15 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  C.  ( A  +P.  B ) )
42 ltprord 8907 . . 3  |-  ( ( A  e.  P.  /\  ( A  +P.  B )  e.  P. )  -> 
( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
435, 42syldan 457 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
4441, 43mpbird 224 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320    C. wpss 3321   (/)c0 3628   class class class wbr 4212    X. cxp 4876  (class class class)co 6081   Q.cnq 8727    +Q cplq 8730    <Q cltq 8733   P.cnp 8734    +P. cpp 8736    <P cltp 8738
This theorem is referenced by:  ltaddpr2  8912  ltexprlem7  8919  ltaprlem  8921  0lt1sr  8970  mappsrpr  8983
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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