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Theorem prplnqu 7440
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 7367 . . . . . . . 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 7417 . . . . . . 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 408 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  +P.  X ) )
7 addcomprg 7398 . . . . . . 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 408 . . . . . 6  |-  ( ph  ->  ( <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  +P.  X
)  =  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
96, 8breqtrd 3954 . . . . 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 7357 . . . . . . . . 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 408 . . . . . . . 8  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )
13 prop 7295 . . . . . . . . 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 7302 . . . . . . . . 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 281 . . . . . . . 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 408 . . . . . . 7  |-  ( ph  ->  A  e.  Q. )
17 nqpru 7372 . . . . . . 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 408 . . . . . 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 146 . . . . 5  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
20 ltsopr 7416 . . . . . 6  |-  <P  Or  P.
21 ltrelpr 7325 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
2220, 21sotri 4934 . . . . 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 408 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
24 ltnqpr 7413 . . . . 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 408 . . . 4  |-  ( ph  ->  ( Q  <Q  A  <->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
2623, 25mpbird 166 . . 3  |-  ( ph  ->  Q  <Q  A )
27 ltexnqi 7229 . . 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 274 . . . . . 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 274 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  Q  e.  Q. )
31 simprl 520 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  Q. )
32 addcomnqg 7201 . . . . . . . . . 10  |-  ( ( Q  e.  Q.  /\  z  e.  Q. )  ->  ( Q  +Q  z
)  =  ( z  +Q  Q ) )
3330, 31, 32syl2anc 408 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  ( z  +Q  Q ) )
34 simprr 521 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  A )
3533, 34eqtr3d 2174 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( z  +Q  Q
)  =  A )
36 breq2 3933 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
l  <Q  ( z  +Q  Q )  <->  l  <Q  A ) )
3736abbidv 2257 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { l  |  l  <Q  (
z  +Q  Q ) }  =  { l  |  l  <Q  A }
)
38 breq1 3932 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
( z  +Q  Q
)  <Q  u  <->  A  <Q  u ) )
3938abbidv 2257 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { u  |  ( z  +Q  Q )  <Q  u }  =  { u  |  A  <Q  u }
)
4037, 39opeq12d 3713 . . . . . . . 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 7381 . . . . . . . 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 408 . . . . . . 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 2174 . . . . . 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 3954 . . . . 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 7439 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
4746adantl 275 . . . . . 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 274 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  e.  P. )
49 nqprlu 7367 . . . . . . 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 7398 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5352adantl 275 . . . . . 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 5940 . . . . 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 166 . . . 4  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
56 nqpru 7372 . . . . 5  |-  ( ( z  e.  Q.  /\  X  e.  P. )  ->  ( z  e.  ( 2nd `  X )  <-> 
X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
)
5731, 48, 56syl2anc 408 . . . 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 166 . . 3  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  ( 2nd `  X ) )
59 oveq1 5781 . . . . 5  |-  ( y  =  z  ->  (
y  +Q  Q )  =  ( z  +Q  Q ) )
6059eqeq1d 2148 . . . 4  |-  ( y  =  z  ->  (
( y  +Q  Q
)  =  A  <->  ( z  +Q  Q )  =  A ) )
6160rspcev 2789 . . 3  |-  ( ( z  e.  ( 2nd `  X )  /\  (
z  +Q  Q )  =  A )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6258, 35, 61syl2anc 408 . 2  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6328, 62rexlimddv 2554 1  |-  ( ph  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331    e. wcel 1480   {cab 2125   E.wrex 2417   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7100    +Q cplq 7102    <Q cltq 7105   P.cnp 7111    +P. cpp 7113    <P cltp 7115
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 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-iplp 7288  df-iltp 7290
This theorem is referenced by:  caucvgprprlemexbt  7526
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