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Theorem ltexprlemfu 7233
Description: Lemma for ltexpri 7235. 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 7127 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4505 . . . . 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 7230 . . . 4  |-  ( A 
<P  B  ->  C  e. 
P. )
6 df-iplp 7090 . . . . 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 6997 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
86, 7genpelvu 7135 . . . 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 404 . . 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 500 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  =  ( w  +Q  u ) )
114ltexprlemelu 7221 . . . . . . . . . . 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 474 . . . . . . . . 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 272 . . . . . . 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 7097 . . . . . . . . . . . . . . 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 7109 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A )  /\  w  e.  ( 2nd `  A
) )  ->  y  <Q  w )
1917, 18syl3an1 1208 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A )  /\  w  e.  ( 2nd `  A
) )  ->  y  <Q  w )
20193com23 1150 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  A )  /\  y  e.  ( 1st `  A
) )  ->  y  <Q  w )
21203adant2r 1170 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) )  /\  y  e.  ( 1st `  A ) )  -> 
y  <Q  w )
22213adant2r 1170 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  y  e.  ( 1st `  A
) )  ->  y  <Q  w )
23223adant3r 1172 . . . . . . . . 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 7022 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2524adantl 272 . . . . . . . . . . 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 7103 . . . . . . . . . . . . . 14  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
2717, 26sylan 278 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  y  e.  ( 1st `  A ) )  -> 
y  e.  Q. )
2827adantrr 464 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  e.  ( 1st `  A )  /\  ( y  +Q  u )  e.  ( 2nd `  B ) ) )  ->  y  e.  Q. )
29283adant2 963 . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  ( 2nd `  A ) )  ->  w  e.  Q. )
3117, 30sylan 278 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  w  e.  ( 2nd `  A ) )  ->  w  e.  Q. )
3231adantrr 464 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  w  e.  Q. )
3332adantrr 464 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  w  e.  Q. )
34333adant3 964 . . . . . . . . . . 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 7097 . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  u  e.  ( 2nd `  C ) )  ->  u  e.  Q. )
3836, 37sylan 278 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  u  e.  ( 2nd `  C ) )  ->  u  e.  Q. )
3938adantrl 463 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  u  e.  Q. )
4039adantrr 464 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  u  e.  Q. )
41403adant3 964 . . . . . . . . . . 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 7003 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
4342adantl 272 . . . . . . . . . . 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 5830 . . . . . . . . . 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 7097 . . . . . . . . . . . . . 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 7107 . . . . . . . . . . . . 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 278 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  u
)  e.  ( 2nd `  B ) )  -> 
( ( y  +Q  u )  <Q  (
w  +Q  u )  ->  ( w  +Q  u )  e.  ( 2nd `  B ) ) )
5049adantrl 463 . . . . . . . . . . 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 963 . . . . . . . . . 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 1144 . . . . . . 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 1827 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( w  +Q  u
)  e.  ( 2nd `  B ) )
5610, 55eqeltrd 2165 . . . . 5  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 2nd `  A
)  /\  u  e.  ( 2nd `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  e.  ( 2nd `  B ) )
5756expr 368 . . . 4  |-  ( ( A  <P  B  /\  ( w  e.  ( 2nd `  A )  /\  u  e.  ( 2nd `  C ) ) )  ->  ( z  =  ( w  +Q  u
)  ->  z  e.  ( 2nd `  B ) ) )
5857rexlimdvva 2499 . . 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 3034 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 925    = wceq 1290   E.wex 1427    e. wcel 1439   E.wrex 2361   {crab 2364    C_ wss 3002   <.cop 3455   class class class wbr 3853   ` cfv 5030  (class class class)co 5668   1stc1st 5925   2ndc2nd 5926   Q.cnq 6902    +Q cplq 6904    <Q cltq 6907   P.cnp 6913    +P. cpp 6915    <P cltp 6917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3962  ax-sep 3965  ax-nul 3973  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-iinf 4418
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2624  df-sbc 2844  df-csb 2937  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-nul 3290  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-int 3697  df-iun 3740  df-br 3854  df-opab 3908  df-mpt 3909  df-tr 3945  df-eprel 4127  df-id 4131  df-po 4134  df-iso 4135  df-iord 4204  df-on 4206  df-suc 4209  df-iom 4421  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467  df-iota 4995  df-fun 5032  df-fn 5033  df-f 5034  df-f1 5035  df-fo 5036  df-f1o 5037  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-1st 5927  df-2nd 5928  df-recs 6086  df-irdg 6151  df-1o 6197  df-2o 6198  df-oadd 6201  df-omul 6202  df-er 6308  df-ec 6310  df-qs 6314  df-ni 6926  df-pli 6927  df-mi 6928  df-lti 6929  df-plpq 6966  df-mpq 6967  df-enq 6969  df-nqqs 6970  df-plqqs 6971  df-mqqs 6972  df-1nqqs 6973  df-rq 6974  df-ltnqqs 6975  df-enq0 7046  df-nq0 7047  df-0nq0 7048  df-plq0 7049  df-mq0 7050  df-inp 7088  df-iplp 7090  df-iltp 7092
This theorem is referenced by:  ltexpri  7235
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