ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltaddpr Unicode version

Theorem ltaddpr 6753
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 6631 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
2 prml 6633 . . . 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 266 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. p  e.  Q.  p  e.  ( 1st `  B ) )
5 prop 6631 . . . . 5  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
6 prarloc 6659 . . . . 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 271 . . . 4  |-  ( ( A  e.  P.  /\  p  e.  Q. )  ->  E. r  e.  ( 1st `  A ) E. q  e.  ( 2nd `  A ) q  <Q  ( r  +Q  p ) )
87ad2ant2r 486 . . 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 6638 . . . . . . . . . . 11  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
105, 9sylan 271 . . . . . . . . . 10  |-  ( ( A  e.  P.  /\  q  e.  ( 2nd `  A ) )  -> 
q  e.  Q. )
1110adantlr 454 . . . . . . . . 9  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  q  e.  ( 2nd `  A ) )  ->  q  e.  Q. )
1211ad2ant2rl 488 . . . . . . . 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 265 . . . . . . 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 496 . . . . . . 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 491 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  r  e.  ( 1st `  A
) )
16 simplr 490 . . . . . . . . . . . . 13  |-  ( ( ( p  e.  Q.  /\  p  e.  ( 1st `  B ) )  /\  ( r  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) )  ->  p  e.  ( 1st `  B
) )
1715, 16jca 294 . . . . . . . . . . . 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 6624 . . . . . . . . . . . . 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 6531 . . . . . . . . . . . . 13  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
2018, 19genpprecll 6670 . . . . . . . . . . . 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 427 . . . . . . . . . 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 6693 . . . . . . . . . . 11  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  +P.  B
)  e.  P. )
24 prop 6631 . . . . . . . . . . . 12  |-  ( ( A  +P.  B )  e.  P.  ->  <. ( 1st `  ( A  +P.  B ) ) ,  ( 2nd `  ( A  +P.  B ) )
>.  e.  P. )
25 prcdnql 6640 . . . . . . . . . . . 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 271 . . . . . . . . . . 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 271 . . . . . . . . . 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 386 . . . . . . . 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 119 . . . . . . 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 2387 . . . . . . 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 1144 . . . . . 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 6662 . . . . . . . 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 270 . . . . . . 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 469 . . . . . 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 160 . . . . 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 112 . . . 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 2457 . . 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 2454 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  A  <P  ( A  +P.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    e. wcel 1409   E.wrex 2324   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  ltexprlemrl  6766  ltaprlem  6774  ltaprg  6775  prplnqu  6776  ltmprr  6798  caucvgprprlemnkltj  6845  caucvgprprlemnkeqj  6846  caucvgprprlemnbj  6849  0lt1sr  6908  recexgt0sr  6916  mulgt0sr  6920  archsr  6924  prsrpos  6927  pitoregt0  6983
  Copyright terms: Public domain W3C validator