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Theorem prplnqu 7610
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 7537 . . . . . . . 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 7587 . . . . . . 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 411 . . . . . 6  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  ( <. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  +P.  X ) )
7 addcomprg 7568 . . . . . . 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 411 . . . . . 6  |-  ( ph  ->  ( <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  +P.  X
)  =  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )
)
96, 8breqtrd 4026 . . . . 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 7527 . . . . . . . . 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 411 . . . . . . . 8  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  e.  P. )
13 prop 7465 . . . . . . . . 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 7472 . . . . . . . . 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 283 . . . . . . . 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 411 . . . . . . 7  |-  ( ph  ->  A  e.  Q. )
17 nqpru 7542 . . . . . . 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 411 . . . . . 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 147 . . . . 5  |-  ( ph  ->  ( X  +P.  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >. )  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
20 ltsopr 7586 . . . . . 6  |-  <P  Or  P.
21 ltrelpr 7495 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
2220, 21sotri 5020 . . . . 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 411 . . . 4  |-  ( ph  -> 
<. { l  |  l 
<Q  Q } ,  {
u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
24 ltnqpr 7583 . . . . 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 411 . . . 4  |-  ( ph  ->  ( Q  <Q  A  <->  <. { l  |  l  <Q  Q } ,  { u  |  Q  <Q  u } >.  <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. )
)
2623, 25mpbird 167 . . 3  |-  ( ph  ->  Q  <Q  A )
27 ltexnqi 7399 . . 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 276 . . . . . 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 276 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  Q  e.  Q. )
31 simprl 529 . . . . . . . . . 10  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  Q. )
32 addcomnqg 7371 . . . . . . . . . 10  |-  ( ( Q  e.  Q.  /\  z  e.  Q. )  ->  ( Q  +Q  z
)  =  ( z  +Q  Q ) )
3330, 31, 32syl2anc 411 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  ( z  +Q  Q ) )
34 simprr 531 . . . . . . . . 9  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( Q  +Q  z
)  =  A )
3533, 34eqtr3d 2212 . . . . . . . 8  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
( z  +Q  Q
)  =  A )
36 breq2 4004 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
l  <Q  ( z  +Q  Q )  <->  l  <Q  A ) )
3736abbidv 2295 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { l  |  l  <Q  (
z  +Q  Q ) }  =  { l  |  l  <Q  A }
)
38 breq1 4003 . . . . . . . . . 10  |-  ( ( z  +Q  Q )  =  A  ->  (
( z  +Q  Q
)  <Q  u  <->  A  <Q  u ) )
3938abbidv 2295 . . . . . . . . 9  |-  ( ( z  +Q  Q )  =  A  ->  { u  |  ( z  +Q  Q )  <Q  u }  =  { u  |  A  <Q  u }
)
4037, 39opeq12d 3784 . . . . . . . 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 7551 . . . . . . . 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 411 . . . . . . 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 2212 . . . . . 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 4026 . . . . 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 7609 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P.  /\  h  e.  P. )  ->  (
f  <P  g  <->  ( h  +P.  f )  <P  (
h  +P.  g )
) )
4746adantl 277 . . . . . 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 276 . . . . . 6  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  e.  P. )
49 nqprlu 7537 . . . . . . 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 7568 . . . . . . 7  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f  +P.  g
)  =  ( g  +P.  f ) )
5352adantl 277 . . . . . 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 6038 . . . . 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 167 . . . 4  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
56 nqpru 7542 . . . . 5  |-  ( ( z  e.  Q.  /\  X  e.  P. )  ->  ( z  e.  ( 2nd `  X )  <-> 
X  <P  <. { l  |  l  <Q  z } ,  { u  |  z 
<Q  u } >. )
)
5731, 48, 56syl2anc 411 . . . 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 167 . . 3  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  -> 
z  e.  ( 2nd `  X ) )
59 oveq1 5876 . . . . 5  |-  ( y  =  z  ->  (
y  +Q  Q )  =  ( z  +Q  Q ) )
6059eqeq1d 2186 . . . 4  |-  ( y  =  z  ->  (
( y  +Q  Q
)  =  A  <->  ( z  +Q  Q )  =  A ) )
6160rspcev 2841 . . 3  |-  ( ( z  e.  ( 2nd `  X )  /\  (
z  +Q  Q )  =  A )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6258, 35, 61syl2anc 411 . 2  |-  ( (
ph  /\  ( z  e.  Q.  /\  ( Q  +Q  z )  =  A ) )  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q )  =  A )
6328, 62rexlimddv 2599 1  |-  ( ph  ->  E. y  e.  ( 2nd `  X ) ( y  +Q  Q
)  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456   <.cop 3594   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    +P. cpp 7283    <P cltp 7285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iplp 7458  df-iltp 7460
This theorem is referenced by:  caucvgprprlemexbt  7696
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