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Theorem ltaddpr 9459
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 9414 . . . . 5  |-  ( B  e.  P.  ->  B  =/=  (/) )
2 n0 3741 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2sylib 200 . . . 4  |-  ( B  e.  P.  ->  E. y 
y  e.  B )
43adantl 468 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. y  y  e.  B )
5 addclpr 9443 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
65adantr 467 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( A  +P.  B )  e. 
P. )
7 df-plp 9408 . . . . . . . . . . . . 13  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
8 addclnq 9370 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
97, 8genpprecl 9426 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) ) )
109imp 431 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) )
11 elprnq 9416 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  +Q  y )  e.  Q. )
12 addnqf 9373 . . . . . . . . . . . . . . 15  |-  +Q  :
( Q.  X.  Q. )
--> Q.
1312fdmi 5734 . . . . . . . . . . . . . 14  |-  dom  +Q  =  ( Q.  X.  Q. )
14 0nnq 9349 . . . . . . . . . . . . . 14  |-  -.  (/)  e.  Q.
1513, 14ndmovrcl 6455 . . . . . . . . . . . . 13  |-  ( ( x  +Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
16 ltaddnq 9399 . . . . . . . . . . . . 13  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  x  <Q  ( x  +Q  y ) )
1711, 15, 163syl 18 . . . . . . . . . . . 12  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  x  <Q  (
x  +Q  y ) )
18 prcdnq 9418 . . . . . . . . . . . 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 667 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  x  e.  ( A  +P.  B
) )
2120exp32 610 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  e.  ( A  +P.  B ) ) ) )
2221com23 81 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  x  e.  ( A  +P.  B ) ) ) )
2322alrimdv 1775 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) ) )
24 dfss2 3421 . . . . . . 7  |-  ( A 
C_  ( A  +P.  B )  <->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) )
2523, 24syl6ibr 231 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C_  ( A  +P.  B ) ) )
26 vex 3048 . . . . . . . . 9  |-  y  e. 
_V
2726prlem934 9458 . . . . . . . 8  |-  ( A  e.  P.  ->  E. x  e.  A  -.  (
x  +Q  y )  e.  A )
2827adantr 467 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. x  e.  A  -.  ( x  +Q  y
)  e.  A )
29 eleq2 2518 . . . . . . . . . . . . 13  |-  ( A  =  ( A  +P.  B )  ->  ( (
x  +Q  y )  e.  A  <->  ( x  +Q  y )  e.  ( A  +P.  B ) ) )
3029biimprcd 229 . . . . . . . . . . . 12  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( A  =  ( A  +P.  B )  ->  ( x  +Q  y )  e.  A
) )
3130con3d 139 . . . . . . . . . . 11  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) )
329, 31syl6 34 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) )
3332expd 438 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( -.  ( x  +Q  y )  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3433com34 86 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( -.  ( x  +Q  y )  e.  A  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3534rexlimdv 2877 . . . . . . 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 536 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B ) ) ) )
38 dfpss2 3518 . . . . 5  |-  ( A 
C.  ( A  +P.  B )  <->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B
) ) )
3937, 38syl6ibr 231 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C.  ( A  +P.  B ) ) )
4039exlimdv 1779 . . 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 9455 . . 3  |-  ( ( A  e.  P.  /\  ( A  +P.  B )  e.  P. )  -> 
( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
435, 42syldan 473 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
4441, 43mpbird 236 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738    C_ wss 3404    C. wpss 3405   (/)c0 3731   class class class wbr 4402    X. cxp 4832  (class class class)co 6290   Q.cnq 9277    +Q cplq 9280    <Q cltq 9283   P.cnp 9284    +P. cpp 9286    <P cltp 9288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-omul 7187  df-er 7363  df-ni 9297  df-pli 9298  df-mi 9299  df-lti 9300  df-plpq 9333  df-mpq 9334  df-ltpq 9335  df-enq 9336  df-nq 9337  df-erq 9338  df-plq 9339  df-mq 9340  df-1nq 9341  df-rq 9342  df-ltnq 9343  df-np 9406  df-plp 9408  df-ltp 9410
This theorem is referenced by:  ltaddpr2  9460  ltexprlem7  9467  ltaprlem  9469  0lt1sr  9519  mappsrpr  9532
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