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Theorem ltaddpr 8654
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 8609 . . . . 5  |-  ( B  e.  P.  ->  B  =/=  (/) )
2 n0 3465 . . . . 5  |-  ( B  =/=  (/)  <->  E. y  y  e.  B )
31, 2sylib 188 . . . 4  |-  ( B  e.  P.  ->  E. y 
y  e.  B )
43adantl 452 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. y  y  e.  B )
5 addclpr 8638 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
65adantr 451 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  ( A  +P.  B )  e. 
P. )
7 df-plp 8603 . . . . . . . . . . . . 13  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
8 addclnq 8565 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
97, 8genpprecl 8621 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) ) )
109imp 418 . . . . . . . . . . 11  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  (
x  +Q  y )  e.  ( A  +P.  B ) )
11 elprnq 8611 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  +Q  y )  e.  Q. )
12 addnqf 8568 . . . . . . . . . . . . . . 15  |-  +Q  :
( Q.  X.  Q. )
--> Q.
1312fdmi 5360 . . . . . . . . . . . . . 14  |-  dom  +Q  =  ( Q.  X.  Q. )
14 0nnq 8544 . . . . . . . . . . . . . 14  |-  -.  (/)  e.  Q.
1513, 14ndmovrcl 5968 . . . . . . . . . . . . 13  |-  ( ( x  +Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
16 ltaddnq 8594 . . . . . . . . . . . . 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 8613 . . . . . . . . . . . 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 14 . . . . . . . . . . 11  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  x  e.  ( A  +P.  B ) )
206, 10, 19syl2anc 642 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( x  e.  A  /\  y  e.  B
) )  ->  x  e.  ( A  +P.  B
) )
2120exp32 588 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  e.  ( A  +P.  B ) ) ) )
2221com23 72 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( x  e.  A  ->  x  e.  ( A  +P.  B ) ) ) )
2322alrimdv 1619 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) ) )
24 dfss2 3170 . . . . . . 7  |-  ( A 
C_  ( A  +P.  B )  <->  A. x ( x  e.  A  ->  x  e.  ( A  +P.  B
) ) )
2523, 24syl6ibr 218 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C_  ( A  +P.  B ) ) )
26 vex 2792 . . . . . . . . 9  |-  y  e. 
_V
2726prlem934 8653 . . . . . . . 8  |-  ( A  e.  P.  ->  E. x  e.  A  -.  (
x  +Q  y )  e.  A )
2827adantr 451 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. x  e.  A  -.  ( x  +Q  y
)  e.  A )
29 eleq2 2345 . . . . . . . . . . . . 13  |-  ( A  =  ( A  +P.  B )  ->  ( (
x  +Q  y )  e.  A  <->  ( x  +Q  y )  e.  ( A  +P.  B ) ) )
3029biimprcd 216 . . . . . . . . . . . 12  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( A  =  ( A  +P.  B )  ->  ( x  +Q  y )  e.  A
) )
3130con3d 125 . . . . . . . . . . 11  |-  ( ( x  +Q  y )  e.  ( A  +P.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) )
329, 31syl6 29 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  ( -.  ( x  +Q  y
)  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) )
3332exp3a 425 . . . . . . . . 9  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( y  e.  B  ->  ( -.  ( x  +Q  y )  e.  A  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3433com34 77 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( x  e.  A  ->  ( -.  ( x  +Q  y )  e.  A  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) ) ) )
3534rexlimdv 2667 . . . . . . 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 14 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  -.  A  =  ( A  +P.  B ) ) )
3725, 36jcad 519 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B ) ) ) )
38 dfpss2 3262 . . . . 5  |-  ( A 
C.  ( A  +P.  B )  <->  ( A  C_  ( A  +P.  B )  /\  -.  A  =  ( A  +P.  B
) ) )
3937, 38syl6ibr 218 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( y  e.  B  ->  A  C.  ( A  +P.  B ) ) )
4039exlimdv 1665 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. y  y  e.  B  ->  A  C.  ( A  +P.  B
) ) )
414, 40mpd 14 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  C.  ( A  +P.  B ) )
42 ltprord 8650 . . 3  |-  ( ( A  e.  P.  /\  ( A  +P.  B )  e.  P. )  -> 
( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
435, 42syldan 456 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  ( A  +P.  B )  <->  A  C.  ( A  +P.  B ) ) )
4441, 43mpbird 223 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 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1685    =/= wne 2447   E.wrex 2545    C_ wss 3153    C. wpss 3154   (/)c0 3456   class class class wbr 4024    X. cxp 4686  (class class class)co 5820   Q.cnq 8470    +Q cplq 8473    <Q cltq 8476   P.cnp 8477    +P. cpp 8479    <P cltp 8481
This theorem is referenced by:  ltaddpr2  8655  ltexprlem7  8662  ltaprlem  8664  0lt1sr  8713  mappsrpr  8726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-omul 6480  df-er 6656  df-ni 8492  df-pli 8493  df-mi 8494  df-lti 8495  df-plpq 8528  df-mpq 8529  df-ltpq 8530  df-enq 8531  df-nq 8532  df-erq 8533  df-plq 8534  df-mq 8535  df-1nq 8536  df-rq 8537  df-ltnq 8538  df-np 8601  df-plp 8603  df-ltp 8605
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