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Theorem ltaddpr 8658
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 8613 . . . . 5  |-  ( B  e.  P.  ->  B  =/=  (/) )
2 n0 3464 . . . . 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 8642 . . . . . . . . . . . 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 8607 . . . . . . . . . . . . 13  |-  +P.  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y  +Q  z ) } )
8 addclnq 8569 . . . . . . . . . . . . 13  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  +Q  z
)  e.  Q. )
97, 8genpprecl 8625 . . . . . . . . . . . 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 8615 . . . . . . . . . . . . 13  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( x  +Q  y
)  e.  ( A  +P.  B ) )  ->  ( x  +Q  y )  e.  Q. )
12 addnqf 8572 . . . . . . . . . . . . . . 15  |-  +Q  :
( Q.  X.  Q. )
--> Q.
1312fdmi 5394 . . . . . . . . . . . . . 14  |-  dom  +Q  =  ( Q.  X.  Q. )
14 0nnq 8548 . . . . . . . . . . . . . 14  |-  -.  (/)  e.  Q.
1513, 14ndmovrcl 6006 . . . . . . . . . . . . 13  |-  ( ( x  +Q  y )  e.  Q.  ->  (
x  e.  Q.  /\  y  e.  Q. )
)
16 ltaddnq 8598 . . . . . . . . . . . . 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 8617 . . . . . . . . . . . 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 3169 . . . . . . 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 2791 . . . . . . . . 9  |-  y  e. 
_V
2726prlem934 8657 . . . . . . . 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 2344 . . . . . . . . . . . . 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 2666 . . . . . . 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 3261 . . . . 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 1664 . . 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 8654 . . 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 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152    C. wpss 3153   (/)c0 3455   class class class wbr 4023    X. cxp 4687  (class class class)co 5858   Q.cnq 8474    +Q cplq 8477    <Q cltq 8480   P.cnp 8481    +P. cpp 8483    <P cltp 8485
This theorem is referenced by:  ltaddpr2  8659  ltexprlem7  8666  ltaprlem  8668  0lt1sr  8717  mappsrpr  8730
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-omul 6484  df-er 6660  df-ni 8496  df-pli 8497  df-mi 8498  df-lti 8499  df-plpq 8532  df-mpq 8533  df-ltpq 8534  df-enq 8535  df-nq 8536  df-erq 8537  df-plq 8538  df-mq 8539  df-1nq 8540  df-rq 8541  df-ltnq 8542  df-np 8605  df-plp 8607  df-ltp 8609
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