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Theorem ltaddpr 7559
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 7437 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prml 7439 . . . 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 7437 . . . . 5  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 prarloc 7465 . . . . 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 506 . . 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 7444 . . . . . . . . . . 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 474 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  ( 2nd `  A ) )  ->  q  e.  Q. )
1211ad2ant2rl 508 . . . . . . . 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 531 . . . . . . 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 526 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  r  e.  ( 1st `  A
) )
16 simplr 525 . . . . . . . . . . . . 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 7430 . . . . . . . . . . . . 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 7337 . . . . . . . . . . . . 13  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
2018, 19genpprecll 7476 . . . . . . . . . . . 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 443 . . . . . . . . . 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 7499 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
24 prop 7437 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
25 prcdnql 7446 . . . . . . . . . . . 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 398 . . . . . . . 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 2519 . . . . . . 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 1231 . . . . . 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 7468 . . . . . . . 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 489 . . . . . 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 2595 . . 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 2592 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 2141   E.wrex 2449   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    +Q cplq 7244    <Q cltq 7247   P.cnp 7253    +P. cpp 7255    <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  ltexprlemrl  7572  ltaprlem  7580  ltaprg  7581  prplnqu  7582  ltmprr  7604  caucvgprprlemnkltj  7651  caucvgprprlemnkeqj  7652  caucvgprprlemnbj  7655  0lt1sr  7727  recexgt0sr  7735  mulgt0sr  7740  archsr  7744  prsrpos  7747  mappsrprg  7766  pitoregt0  7811
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