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Theorem prplnqu 6872
Description: Membership in the upper cut of a sum of a positive real and a fraction. (Contributed by Jim Kingdon, 16-Jun-2021.)
Hypotheses
Ref Expression
prplnqu.x  |-  ( ph  ->  X  e.  P. )
prplnqu.q  |-  ( ph  ->  Q  e.  Q. )
prplnqu.sum  |-  ( ph  ->  A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
Assertion
Ref Expression
prplnqu  |-  ( ph  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q
)  =  A )
Distinct variable groups:    A, l, u   
y, A    Q, l, u    y, Q    y, X
Allowed substitution hints:    ph( y, u, l)    X( u, l)

Proof of Theorem prplnqu
Dummy variables  f  g  h  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prplnqu.q . . . . . . . 8  |-  ( ph  ->  Q  e.  Q. )
2 nqprlu 6799 . . . . . . . 8  |-  ( Q  e.  Q.  ->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  e.  P. )
31, 2syl 14 . . . . . . 7  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )
4 prplnqu.x . . . . . . 7  |-  ( ph  ->  X  e.  P. )
5 ltaddpr 6849 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P.  /\  X  e.  P. )  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  +P.  X ) )
63, 4, 5syl2anc 403 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  +P.  X ) )
7 addcomprg 6830 . . . . . . 7  |-  ( (
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P.  /\  X  e.  P. )  ->  ( <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  +P.  X
)  =  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
83, 4, 7syl2anc 403 . . . . . 6  |-  ( ph  ->  ( <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  +P.  X
)  =  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
96, 8breqtrd 3817 . . . . 5  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
10 prplnqu.sum . . . . . 6  |-  ( ph  ->  A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )
11 addclpr 6789 . . . . . . . . 9  |-  ( ( X  e.  P.  /\  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )  ->  ( X  +P.  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >. )  e.  P. )
124, 3, 11syl2anc 403 . . . . . . . 8  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )
13 prop 6727 . . . . . . . . 9  |-  ( ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P.  ->  <. ( 1st `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) ,  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) >.  e.  P. )
14 elprnqu 6734 . . . . . . . . 9  |-  ( (
<. ( 1st `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) ,  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) >.  e.  P.  /\  A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )  ->  A  e.  Q. )
1513, 14sylan 277 . . . . . . . 8  |-  ( ( ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P.  /\  A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) ) )  ->  A  e.  Q. )
1612, 10, 15syl2anc 403 . . . . . . 7  |-  ( ph  ->  A  e.  Q. )
17 nqpru 6804 . . . . . . 7  |-  ( ( A  e.  Q.  /\  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )  ->  ( A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. ) )  <->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P 
<. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >. ) )
1816, 12, 17syl2anc 403 . . . . . 6  |-  ( ph  ->  ( A  e.  ( 2nd `  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)  <->  ( X  +P.  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
1910, 18mpbid 145 . . . . 5  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
20 ltsopr 6848 . . . . . 6  |-  <P  Or  P.
21 ltrelpr 6757 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
2220, 21sotri 4750 . . . . 5  |-  ( (
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  /\  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
239, 19, 22syl2anc 403 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
24 ltnqpr 6845 . . . . 5  |-  ( ( Q  e.  Q.  /\  A  e.  Q. )  ->  ( Q  <Q  A  <->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
251, 16, 24syl2anc 403 . . . 4  |-  ( ph  ->  ( Q  <Q  A  <->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
2623, 25mpbird 165 . . 3  |-  ( ph  ->  Q  <Q  A )
27 ltexnqi 6661 . . 3  |-  ( Q 
<Q  A  ->  E. z  e.  Q.  ( Q  +Q  z )  =  A )
2826, 27syl 14 . 2  |-  ( ph  ->  E. z  e.  Q.  ( Q  +Q  z
)  =  A )
2919adantr 270 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
301adantr 270 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  Q  e.  Q. )
31 simprl 498 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  Q. )
32 addcomnqg 6633 . . . . . . . . . 10  |-  ( ( Q  e.  Q.  /\  z  e.  Q. )  ->  ( Q  +Q  z
)  =  ( z  +Q  Q ) )
3330, 31, 32syl2anc 403 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  ( z  +Q  Q ) )
34 simprr 499 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  A )
3533, 34eqtr3d 2116 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( z  +Q  Q
)  =  A )
36 breq2 3797 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
l  <Q  ( z  +Q  Q )  <->  l  <Q  A ) )
3736abbidv 2197 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { l  |  l  <Q  (
z  +Q  Q ) }  =  { l  |  l  <Q  A }
)
38 breq1 3796 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
( z  +Q  Q
)  <Q  u  <->  A  <Q  u ) )
3938abbidv 2197 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { u  |  ( z  +Q  Q )  <Q  u }  =  { u  |  A  <Q  u }
)
4037, 39opeq12d 3586 . . . . . . . 8  |-  ( ( z  +Q  Q )  =  A  ->  <. { l  |  l  <Q  (
z  +Q  Q ) } ,  { u  |  ( z  +Q  Q )  <Q  u } >.  =  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
4135, 40syl 14 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  <. { l  |  l 
<Q  ( z  +Q  Q
) } ,  {
u  |  ( z  +Q  Q )  <Q  u } >.  =  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
42 addnqpr 6813 . . . . . . . 8  |-  ( ( z  e.  Q.  /\  Q  e.  Q. )  -> 
<. { l  |  l 
<Q  ( z  +Q  Q
) } ,  {
u  |  ( z  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  z } ,  {
u  |  z  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
4331, 30, 42syl2anc 403 . . . . . . 7  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  <. { l  |  l 
<Q  ( z  +Q  Q
) } ,  {
u  |  ( z  +Q  Q )  <Q  u } >.  =  ( <. { l  |  l 
<Q  z } ,  {
u  |  z  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
4441, 43eqtr3d 2116 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  <. { l  |  l 
<Q  A } ,  {
u  |  A  <Q  u } >.  =  ( <. { l  |  l 
<Q  z } ,  {
u  |  z  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
4529, 44breqtrd 3817 . . . . 5  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( <. { l  |  l  <Q 
z } ,  {
u  |  z  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
46 ltaprg 6871 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
4746adantl 271 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  Q.  /\  ( Q  +Q  z
)  =  A ) )  /\  ( f  e.  P.  /\  g  e.  P.  /\  h  e. 
P. ) )  -> 
( f  <P  g  <->  ( h  +P.  f ) 
<P  ( h  +P.  g
) ) )
484adantr 270 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  e.  P. )
49 nqprlu 6799 . . . . . . 7  |-  ( z  e.  Q.  ->  <. { l  |  l  <Q  z } ,  { u  |  z  <Q  u } >.  e.  P. )
5031, 49syl 14 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  <. { l  |  l 
<Q  z } ,  {
u  |  z  <Q  u } >.  e.  P. )
5130, 2syl 14 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  e.  P. )
52 addcomprg 6830 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5352adantl 271 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  Q.  /\  ( Q  +Q  z
)  =  A ) )  /\  ( f  e.  P.  /\  g  e.  P. ) )  -> 
( f  +P.  g
)  =  ( g  +P.  f ) )
5447, 48, 50, 51, 53caovord2d 5701 . . . . 5  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( X  <P  <. { l  |  l  <Q  z } ,  { u  |  z  <Q  u } >.  <-> 
( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  ( <. { l  |  l  <Q 
z } ,  {
u  |  z  <Q  u } >.  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
) )
5545, 54mpbird 165 . . . 4  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
56 nqpru 6804 . . . . 5  |-  ( ( z  e.  Q.  /\  X  e.  P. )  ->  ( z  e.  ( 2nd `  X )  <-> 
X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
)
5731, 48, 56syl2anc 403 . . . 4  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( z  e.  ( 2nd `  X )  <-> 
X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
)
5855, 57mpbird 165 . . 3  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  ( 2nd `  X ) )
59 oveq1 5550 . . . . 5  |-  ( y  =  z  ->  (
y  +Q  Q )  =  ( z  +Q  Q ) )
6059eqeq1d 2090 . . . 4  |-  ( y  =  z  ->  (
( y  +Q  Q
)  =  A  <->  ( z  +Q  Q )  =  A ) )
6160rspcev 2702 . . 3  |-  ( ( z  e.  ( 2nd `  X )  /\  (
z  +Q  Q )  =  A )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6258, 35, 61syl2anc 403 . 2  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6328, 62rexlimddv 2482 1  |-  ( ph  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 920    = wceq 1285    e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3409   class class class wbr 3793   ` cfv 4932  (class class class)co 5543   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532    +Q cplq 6534    <Q cltq 6537   P.cnp 6543    +P. cpp 6545    <P cltp 6547
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-2o 6066  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605  df-enq0 6676  df-nq0 6677  df-0nq0 6678  df-plq0 6679  df-mq0 6680  df-inp 6718  df-iplp 6720  df-iltp 6722
This theorem is referenced by:  caucvgprprlemexbt  6958
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