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Theorem ltexprlemfu 7426
Description: Lemma for ltexpri 7428. One direction of our result for upper cuts. (Contributed by Jim Kingdon, 17-Dec-2019.)
Hypothesis
Ref Expression
ltexprlem.1  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
Assertion
Ref Expression
ltexprlemfu  |-  ( A 
<P  B  ->  ( 2nd `  ( A  +P.  C
) )  C_  ( 2nd `  B ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem ltexprlemfu
Dummy variables  z  w  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7320 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4591 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 111 . . . 4  |-  ( A 
<P  B  ->  A  e. 
P. )
4 ltexprlem.1 . . . . 5  |-  C  = 
<. { x  e.  Q.  |  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  x )  e.  ( 1st `  B ) ) } ,  {
x  e.  Q.  |  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  x )  e.  ( 2nd `  B ) ) } >.
54ltexprlempr 7423 . . . 4  |-  ( A 
<P  B  ->  C  e. 
P. )
6 df-iplp 7283 . . . . 5  |-  +P.  =  ( z  e.  P. ,  y  e.  P.  |->  <. { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 1st `  z )  /\  h  e.  ( 1st `  y
)  /\  f  =  ( g  +Q  h
) ) } ,  { f  e.  Q.  |  E. g  e.  Q.  E. h  e.  Q.  (
g  e.  ( 2nd `  z )  /\  h  e.  ( 2nd `  y
)  /\  f  =  ( g  +Q  h
) ) } >. )
7 addclnq 7190 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
86, 7genpelvu 7328 . . . 4  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( z  e.  ( 2nd `  ( A  +P.  C ) )  <->  E. w  e.  ( 2nd `  A ) E. u  e.  ( 2nd `  C ) z  =  ( w  +Q  u
) ) )
93, 5, 8syl2anc 408 . . 3  |-  ( A 
<P  B  ->  ( z  e.  ( 2nd `  ( A  +P.  C ) )  <->  E. w  e.  ( 2nd `  A ) E. u  e.  ( 2nd `  C ) z  =  ( w  +Q  u
) ) )
10 simprr 521 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  =  ( w  +Q  u ) )
114ltexprlemelu 7414 . . . . . . . . . . 11  |-  ( u  e.  ( 2nd `  C
)  <->  ( u  e. 
Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) ) )
1211biimpi 119 . . . . . . . . . 10  |-  ( u  e.  ( 2nd `  C
)  ->  ( u  e.  Q.  /\  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) ) )
1312ad2antlr 480 . . . . . . . . 9  |-  ( ( ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  ->  (
u  e.  Q.  /\  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) ) )
1413simprd 113 . . . . . . . 8  |-  ( ( ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  ->  E. y
( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )
1514adantl 275 . . . . . . 7  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  E. y ( y  e.  ( 1st `  A
)  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )
16 prop 7290 . . . . . . . . . . . . . . 15  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
173, 16syl 14 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
18 prltlu 7302 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  w  e.  ( 2nd `  A
) )  ->  y  <Q  w )
1917, 18syl3an1 1249 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A )  /\  w  e.  ( 2nd `  A
) )  ->  y  <Q  w )
20193com23 1187 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  A )  /\  y  e.  ( 1st `  A
) )  ->  y  <Q  w )
21203adant2r 1211 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  <Q  w )
22213adant2r 1211 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  y  e.  ( 1st `  A
) )  ->  y  <Q  w )
23223adant3r 1213 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
y  <Q  w )
24 ltanqg 7215 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2524adantl 275 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
26 elprnql 7296 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
2717, 26sylan 281 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
2827adantrr 470 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )  ->  y  e.  Q. )
29283adant2 1000 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
y  e.  Q. )
30 elprnqu 7297 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  ( 2nd `  A ) )  ->  w  e.  Q. )
3117, 30sylan 281 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  A ) )  ->  w  e.  Q. )
3231adantrr 470 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  w  e.  Q. )
3332adantrr 470 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  w  e.  Q. )
34333adant3 1001 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  ->  w  e.  Q. )
35 prop 7290 . . . . . . . . . . . . . . . 16  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
365, 35syl 14 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
37 elprnqu 7297 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  u  e.  ( 2nd `  C ) )  ->  u  e.  Q. )
3836, 37sylan 281 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  C ) )  ->  u  e.  Q. )
3938adantrl 469 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  u  e.  Q. )
4039adantrr 470 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  u  e.  Q. )
41403adant3 1001 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  ->  u  e.  Q. )
42 addcomnqg 7196 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 275 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
4425, 29, 34, 41, 43caovord2d 5940 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
( y  <Q  w  <->  ( y  +Q  u ) 
<Q  ( w  +Q  u
) ) )
452simprd 113 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
46 prop 7290 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
4745, 46syl 14 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
48 prcunqu 7300 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  u
)  e.  ( 2nd `  B ) )  -> 
( ( y  +Q  u )  <Q  (
w  +Q  u )  ->  ( w  +Q  u )  e.  ( 2nd `  B ) ) )
4947, 48sylan 281 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  u
)  e.  ( 2nd `  B ) )  -> 
( ( y  +Q  u )  <Q  (
w  +Q  u )  ->  ( w  +Q  u )  e.  ( 2nd `  B ) ) )
5049adantrl 469 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )  ->  (
( y  +Q  u
)  <Q  ( w  +Q  u )  ->  (
w  +Q  u )  e.  ( 2nd `  B
) ) )
51503adant2 1000 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
( ( y  +Q  u )  <Q  (
w  +Q  u )  ->  ( w  +Q  u )  e.  ( 2nd `  B ) ) )
5244, 51sylbid 149 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
( y  <Q  w  ->  ( w  +Q  u
)  e.  ( 2nd `  B ) ) )
5323, 52mpd 13 . . . . . . . 8  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 1st `  A )  /\  (
y  +Q  u )  e.  ( 2nd `  B
) ) )  -> 
( w  +Q  u
)  e.  ( 2nd `  B ) )
54533expa 1181 . . . . . . 7  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )  ->  (
w  +Q  u )  e.  ( 2nd `  B
) )
5515, 54exlimddv 1870 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( w  +Q  u
)  e.  ( 2nd `  B ) )
5610, 55eqeltrd 2216 . . . . 5  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  e.  ( 2nd `  B ) )
5756expr 372 . . . 4  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  ( z  =  ( w  +Q  u
)  ->  z  e.  ( 2nd `  B ) ) )
5857rexlimdvva 2557 . . 3  |-  ( A 
<P  B  ->  ( E. w  e.  ( 2nd `  A ) E. u  e.  ( 2nd `  C
) z  =  ( w  +Q  u )  ->  z  e.  ( 2nd `  B ) ) )
599, 58sylbid 149 . 2  |-  ( A 
<P  B  ->  ( z  e.  ( 2nd `  ( A  +P.  C ) )  ->  z  e.  ( 2nd `  B ) ) )
6059ssrdv 3103 1  |-  ( A 
<P  B  ->  ( 2nd `  ( A  +P.  C
) )  C_  ( 2nd `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 962    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2417   {crab 2420    C_ wss 3071   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095    +Q cplq 7097    <Q cltq 7100   P.cnp 7106    +P. cpp 7108    <P cltp 7110
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 7119  df-pli 7120  df-mi 7121  df-lti 7122  df-plpq 7159  df-mpq 7160  df-enq 7162  df-nqqs 7163  df-plqqs 7164  df-mqqs 7165  df-1nqqs 7166  df-rq 7167  df-ltnqqs 7168  df-enq0 7239  df-nq0 7240  df-0nq0 7241  df-plq0 7242  df-mq0 7243  df-inp 7281  df-iplp 7283  df-iltp 7285
This theorem is referenced by:  ltexpri  7428
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