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Theorem ltaddpr 8942
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 8897 . . . . 5  |-  ( B  e.  P.  ->  B  =/=  (/) )
2 n0 3622 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2sylib 190 . . . 4  |-  ( B  e.  P.  ->  E. y 
y  e.  B )
43adantl 454 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. y  y  e.  B )
5 addclpr 8926 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
65adantr 453 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( A  +P.  B )  e. 
P. )
7 df-plp 8891 . . . . . . . . . . . . 13  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
8 addclnq 8853 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
97, 8genpprecl 8909 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) ) )
109imp 420 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) )
11 elprnq 8899 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  +Q  y )  e.  Q. )
12 addnqf 8856 . . . . . . . . . . . . . . 15  |-  +Q  :
( Q.  X.  Q. )
--> Q.
1312fdmi 5625 . . . . . . . . . . . . . 14  |-  dom  +Q  =  ( Q.  X.  Q. )
14 0nnq 8832 . . . . . . . . . . . . . 14  |-  -.  (/)  e.  Q.
1513, 14ndmovrcl 6262 . . . . . . . . . . . . 13  |-  ( ( x  +Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
16 ltaddnq 8882 . . . . . . . . . . . . 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 8901 . . . . . . . . . . . 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 644 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  x  e.  ( A  +P.  B
) )
2120exp32 590 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  e.  ( A  +P.  B ) ) ) )
2221com23 75 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  x  e.  ( A  +P.  B ) ) ) )
2322alrimdv 1644 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) ) )
24 dfss2 3323 . . . . . . 7  |-  ( A 
C_  ( A  +P.  B )  <->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) )
2523, 24syl6ibr 220 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C_  ( A  +P.  B ) ) )
26 vex 2965 . . . . . . . . 9  |-  y  e. 
_V
2726prlem934 8941 . . . . . . . 8  |-  ( A  e.  P.  ->  E. x  e.  A  -.  (
x  +Q  y )  e.  A )
2827adantr 453 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. x  e.  A  -.  ( x  +Q  y
)  e.  A )
29 eleq2 2503 . . . . . . . . . . . . 13  |-  ( A  =  ( A  +P.  B )  ->  ( (
x  +Q  y )  e.  A  <->  ( x  +Q  y )  e.  ( A  +P.  B ) ) )
3029biimprcd 218 . . . . . . . . . . . 12  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( A  =  ( A  +P.  B )  ->  ( x  +Q  y )  e.  A
) )
3130con3d 128 . . . . . . . . . . 11  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) )
329, 31syl6 32 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) )
3332exp3a 427 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( -.  ( x  +Q  y )  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3433com34 80 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( -.  ( x  +Q  y )  e.  A  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3534rexlimdv 2835 . . . . . . 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 521 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B ) ) ) )
38 dfpss2 3418 . . . . 5  |-  ( A 
C.  ( A  +P.  B )  <->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B
) ) )
3937, 38syl6ibr 220 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C.  ( A  +P.  B ) ) )
4039exlimdv 1647 . . 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 8938 . . 3  |-  ( ( A  e.  P.  /\  ( A  +P.  B )  e.  P. )  -> 
( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
435, 42syldan 458 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
4441, 43mpbird 225 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 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712    C_ wss 3306    C. wpss 3307   (/)c0 3613   class class class wbr 4237    X. cxp 4905  (class class class)co 6110   Q.cnq 8758    +Q cplq 8761    <Q cltq 8764   P.cnp 8765    +P. cpp 8767    <P cltp 8769
This theorem is referenced by:  ltaddpr2  8943  ltexprlem7  8950  ltaprlem  8952  0lt1sr  9001  mappsrpr  9014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-omul 6758  df-er 6934  df-ni 8780  df-pli 8781  df-mi 8782  df-lti 8783  df-plpq 8816  df-mpq 8817  df-ltpq 8818  df-enq 8819  df-nq 8820  df-erq 8821  df-plq 8822  df-mq 8823  df-1nq 8824  df-rq 8825  df-ltnq 8826  df-np 8889  df-plp 8891  df-ltp 8893
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