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Theorem ltaddpr 7419
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.)
Assertion
Ref Expression
ltaddpr  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )

Proof of Theorem ltaddpr
Dummy variables  f  g  h  x  y  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7297 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prml 7299 . . . 4  |-  ( <.
( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  ->  E. p  e.  Q.  p  e.  ( 1st `  B ) )
31, 2syl 14 . . 3  |-  ( B  e.  P.  ->  E. p  e.  Q.  p  e.  ( 1st `  B ) )
43adantl 275 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. p  e.  Q.  p  e.  ( 1st `  B ) )
5 prop 7297 . . . . 5  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 prarloc 7325 . . . . 5  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  p  e.  Q. )  ->  E. r  e.  ( 1st `  A ) E. q  e.  ( 2nd `  A ) q  <Q  ( r  +Q  p ) )
75, 6sylan 281 . . . 4  |-  ( ( A  e.  P.  /\  p  e.  Q. )  ->  E. r  e.  ( 1st `  A ) E. q  e.  ( 2nd `  A ) q  <Q  ( r  +Q  p ) )
87ad2ant2r 500 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  ->  E. r  e.  ( 1st `  A ) E. q  e.  ( 2nd `  A ) q  <Q  ( r  +Q  p ) )
9 elprnqu 7304 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
105, 9sylan 281 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
1110adantlr 468 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  ( 2nd `  A ) )  ->  q  e.  Q. )
1211ad2ant2rl 502 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  ->  q  e.  Q. )
1312adantr 274 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  q  e.  Q. )
14 simplrr 525 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  q  e.  ( 2nd `  A
) )
15 simprl 520 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  r  e.  ( 1st `  A
) )
16 simplr 519 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  p  e.  ( 1st `  B
) )
1715, 16jca 304 . . . . . . . . . . . 12  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  (
r  e.  ( 1st `  A )  /\  p  e.  ( 1st `  B
) ) )
18 df-iplp 7290 . . . . . . . . . . . . 13  |-  +P.  =  ( x  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  x )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  x )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
19 addclnq 7197 . . . . . . . . . . . . 13  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
2018, 19genpprecll 7336 . . . . . . . . . . . 12  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( r  e.  ( 1st `  A
)  /\  p  e.  ( 1st `  B ) )  ->  ( r  +Q  p )  e.  ( 1st `  ( A  +P.  B ) ) ) )
2117, 20syl5 32 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B
) )  /\  (
r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A
) ) )  -> 
( r  +Q  p
)  e.  ( 1st `  ( A  +P.  B
) ) ) )
2221imdistani 441 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( p  e. 
Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) ) )  -> 
( ( A  e. 
P.  /\  B  e.  P. )  /\  (
r  +Q  p )  e.  ( 1st `  ( A  +P.  B ) ) ) )
23 addclpr 7359 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
24 prop 7297 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
25 prcdnql 7306 . . . . . . . . . . . 12  |-  ( (
<. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P.  /\  (
r  +Q  p )  e.  ( 1st `  ( A  +P.  B ) ) )  ->  ( q  <Q  ( r  +Q  p
)  ->  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
2624, 25sylan 281 . . . . . . . . . . 11  |-  ( ( ( A  +P.  B
)  e.  P.  /\  ( r  +Q  p
)  e.  ( 1st `  ( A  +P.  B
) ) )  -> 
( q  <Q  (
r  +Q  p )  ->  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
2723, 26sylan 281 . . . . . . . . . 10  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( r  +Q  p
)  e.  ( 1st `  ( A  +P.  B
) ) )  -> 
( q  <Q  (
r  +Q  p )  ->  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
2822, 27syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( ( p  e. 
Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) ) )  -> 
( q  <Q  (
r  +Q  p )  ->  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
2928anassrs 397 . . . . . . . 8  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  ->  (
q  <Q  ( r  +Q  p )  ->  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
3029imp 123 . . . . . . 7  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  q  e.  ( 1st `  ( A  +P.  B ) ) )
31 rspe 2481 . . . . . . 7  |-  ( ( q  e.  Q.  /\  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) )  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
3213, 14, 30, 31syl12anc 1214 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) )
33 ltdfpr 7328 . . . . . . . 8  |-  ( ( A  e.  P.  /\  ( A  +P.  B )  e.  P. )  -> 
( A  <P  ( A  +P.  B )  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) ) )
3423, 33syldan 280 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  ( A  +P.  B )  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A
)  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) ) )
3534ad3antrrr 483 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  ( A  <P  ( A  +P.  B )  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  ( A  +P.  B ) ) ) ) )
3632, 35mpbird 166 . . . . 5  |-  ( ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  /\  q  <Q  ( r  +Q  p
) )  ->  A  <P  ( A  +P.  B
) )
3736ex 114 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  B  e.  P. )  /\  (
p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  /\  ( r  e.  ( 1st `  A
)  /\  q  e.  ( 2nd `  A ) ) )  ->  (
q  <Q  ( r  +Q  p )  ->  A  <P  ( A  +P.  B
) ) )
3837rexlimdvva 2557 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  ->  ( E. r  e.  ( 1st `  A
) E. q  e.  ( 2nd `  A
) q  <Q  (
r  +Q  p )  ->  A  <P  ( A  +P.  B ) ) )
398, 38mpd 13 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( p  e.  Q.  /\  p  e.  ( 1st `  B ) ) )  ->  A  <P  ( A  +P.  B ) )
404, 39rexlimddv 2554 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1480   E.wrex 2417   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7102    +Q cplq 7104    <Q cltq 7107   P.cnp 7113    +P. cpp 7115    <P cltp 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7126  df-pli 7127  df-mi 7128  df-lti 7129  df-plpq 7166  df-mpq 7167  df-enq 7169  df-nqqs 7170  df-plqqs 7171  df-mqqs 7172  df-1nqqs 7173  df-rq 7174  df-ltnqqs 7175  df-enq0 7246  df-nq0 7247  df-0nq0 7248  df-plq0 7249  df-mq0 7250  df-inp 7288  df-iplp 7290  df-iltp 7292
This theorem is referenced by:  ltexprlemrl  7432  ltaprlem  7440  ltaprg  7441  prplnqu  7442  ltmprr  7464  caucvgprprlemnkltj  7511  caucvgprprlemnkeqj  7512  caucvgprprlemnbj  7515  0lt1sr  7587  recexgt0sr  7595  mulgt0sr  7600  archsr  7604  prsrpos  7607  mappsrprg  7626  pitoregt0  7671
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