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Theorem ltaddpr 7135
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 7013 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prml 7015 . . . 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 271 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. p  e.  Q.  p  e.  ( 1st `  B ) )
5 prop 7013 . . . . 5  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 prarloc 7041 . . . . 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 277 . . . 4  |-  ( ( A  e.  P.  /\  p  e.  Q. )  ->  E. r  e.  ( 1st `  A ) E. q  e.  ( 2nd `  A ) q  <Q  ( r  +Q  p ) )
87ad2ant2r 493 . . 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 7020 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
105, 9sylan 277 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
1110adantlr 461 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  ( 2nd `  A ) )  ->  q  e.  Q. )
1211ad2ant2rl 495 . . . . . . . 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 270 . . . . . . 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 503 . . . . . . 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 498 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  r  e.  ( 1st `  A
) )
16 simplr 497 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  p  e.  ( 1st `  B
) )
1715, 16jca 300 . . . . . . . . . . . 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 7006 . . . . . . . . . . . . 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 6913 . . . . . . . . . . . . 13  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
2018, 19genpprecll 7052 . . . . . . . . . . . 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 434 . . . . . . . . . 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 7075 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
24 prop 7013 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
25 prcdnql 7022 . . . . . . . . . . . 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 277 . . . . . . . . . . 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 277 . . . . . . . . . 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 392 . . . . . . . 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 122 . . . . . . 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 2424 . . . . . . 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 1172 . . . . . 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 7044 . . . . . . . 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 276 . . . . . . 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 476 . . . . . 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 165 . . . . 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 113 . . . 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 2496 . . 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 2493 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1438   E.wrex 2360   <.cop 3444   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   1stc1st 5891   2ndc2nd 5892   Q.cnq 6818    +Q cplq 6820    <Q cltq 6823   P.cnp 6829    +P. cpp 6831    <P cltp 6833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-eprel 4107  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-irdg 6117  df-1o 6163  df-2o 6164  df-oadd 6167  df-omul 6168  df-er 6272  df-ec 6274  df-qs 6278  df-ni 6842  df-pli 6843  df-mi 6844  df-lti 6845  df-plpq 6882  df-mpq 6883  df-enq 6885  df-nqqs 6886  df-plqqs 6887  df-mqqs 6888  df-1nqqs 6889  df-rq 6890  df-ltnqqs 6891  df-enq0 6962  df-nq0 6963  df-0nq0 6964  df-plq0 6965  df-mq0 6966  df-inp 7004  df-iplp 7006  df-iltp 7008
This theorem is referenced by:  ltexprlemrl  7148  ltaprlem  7156  ltaprg  7157  prplnqu  7158  ltmprr  7180  caucvgprprlemnkltj  7227  caucvgprprlemnkeqj  7228  caucvgprprlemnbj  7231  0lt1sr  7290  recexgt0sr  7298  mulgt0sr  7302  archsr  7306  prsrpos  7309  pitoregt0  7365
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