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Theorem ltexprlemfl 7229
Description: Lemma for ltexpri 7233. One directon of our result for lower 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
ltexprlemfl  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  C
) )  C_  ( 1st `  B ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y

Proof of Theorem ltexprlemfl
Dummy variables  z  w  u  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltrelpr 7125 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4503 . . . . 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 7228 . . . 4  |-  ( A 
<P  B  ->  C  e. 
P. )
6 df-iplp 7088 . . . . 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 6995 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
86, 7genpelvl 7132 . . . 4  |-  ( ( A  e.  P.  /\  C  e.  P. )  ->  ( z  e.  ( 1st `  ( A  +P.  C ) )  <->  E. w  e.  ( 1st `  A ) E. u  e.  ( 1st `  C ) z  =  ( w  +Q  u
) ) )
93, 5, 8syl2anc 404 . . 3  |-  ( A 
<P  B  ->  ( z  e.  ( 1st `  ( A  +P.  C ) )  <->  E. w  e.  ( 1st `  A ) E. u  e.  ( 1st `  C ) z  =  ( w  +Q  u
) ) )
10 simprr 500 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  =  ( w  +Q  u ) )
114ltexprlemell 7218 . . . . . . . . . . 11  |-  ( u  e.  ( 1st `  C
)  <->  ( u  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1211biimpi 119 . . . . . . . . . 10  |-  ( u  e.  ( 1st `  C
)  ->  ( u  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1312ad2antlr 474 . . . . . . . . 9  |-  ( ( ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  ->  (
u  e.  Q.  /\  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1413adantl 272 . . . . . . . 8  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( u  e.  Q.  /\ 
E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1514simprd 113 . . . . . . 7  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  E. y ( y  e.  ( 2nd `  A
)  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) )
16 prop 7095 . . . . . . . . . . . . . 14  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
173, 16syl 14 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
18 prltlu 7107 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
1917, 18syl3an1 1208 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
20193adant2r 1170 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) )  /\  y  e.  ( 2nd `  A ) )  ->  w  <Q  y )
21203adant2r 1170 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
22213adant3r 1172 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  ->  w  <Q  y )
23 ltanqg 7020 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2423adantl 272 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
25 ltrelnq 6985 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
2625brel 4503 . . . . . . . . . . . . 13  |-  ( w 
<Q  y  ->  ( w  e.  Q.  /\  y  e.  Q. ) )
2722, 26syl 14 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
( w  e.  Q.  /\  y  e.  Q. )
)
2827simpld 111 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  ->  w  e.  Q. )
2927simprd 113 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
y  e.  Q. )
30 prop 7095 . . . . . . . . . . . . . . . 16  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
315, 30syl 14 . . . . . . . . . . . . . . 15  |-  ( A 
<P  B  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
32 elprnql 7101 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3331, 32sylan 278 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3433adantrl 463 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  u  e.  Q. )
3534adantrr 464 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  u  e.  Q. )
36353adant3 964 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  ->  u  e.  Q. )
37 addcomnqg 7001 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3837adantl 272 . . . . . . . . . . 11  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  /\  ( f  e.  Q.  /\  g  e.  Q. )
)  ->  ( f  +Q  g )  =  ( g  +Q  f ) )
3924, 28, 29, 36, 38caovord2d 5828 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
( w  <Q  y  <->  ( w  +Q  u ) 
<Q  ( y  +Q  u
) ) )
402simprd 113 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
41 prop 7095 . . . . . . . . . . . . . 14  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
4240, 41syl 14 . . . . . . . . . . . . 13  |-  ( A 
<P  B  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
43 prcdnql 7104 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  ( y  +Q  u
)  e.  ( 1st `  B ) )  -> 
( ( w  +Q  u )  <Q  (
y  +Q  u )  ->  ( w  +Q  u )  e.  ( 1st `  B ) ) )
4442, 43sylan 278 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  u
)  e.  ( 1st `  B ) )  -> 
( ( w  +Q  u )  <Q  (
y  +Q  u )  ->  ( w  +Q  u )  e.  ( 1st `  B ) ) )
4544adantrl 463 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) )  ->  (
( w  +Q  u
)  <Q  ( y  +Q  u )  ->  (
w  +Q  u )  e.  ( 1st `  B
) ) )
46453adant2 963 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
( ( w  +Q  u )  <Q  (
y  +Q  u )  ->  ( w  +Q  u )  e.  ( 1st `  B ) ) )
4739, 46sylbid 149 . . . . . . . . 9  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
( w  <Q  y  ->  ( w  +Q  u
)  e.  ( 1st `  B ) ) )
4822, 47mpd 13 . . . . . . . 8  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  (
y  e.  ( 2nd `  A )  /\  (
y  +Q  u )  e.  ( 1st `  B
) ) )  -> 
( w  +Q  u
)  e.  ( 1st `  B ) )
49483expa 1144 . . . . . . 7  |-  ( ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  /\  ( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) )  ->  (
w  +Q  u )  e.  ( 1st `  B
) )
5015, 49exlimddv 1827 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( w  +Q  u
)  e.  ( 1st `  B ) )
5110, 50eqeltrd 2165 . . . . 5  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  e.  ( 1st `  B ) )
5251expr 368 . . . 4  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  ( z  =  ( w  +Q  u
)  ->  z  e.  ( 1st `  B ) ) )
5352rexlimdvva 2497 . . 3  |-  ( A 
<P  B  ->  ( E. w  e.  ( 1st `  A ) E. u  e.  ( 1st `  C
) z  =  ( w  +Q  u )  ->  z  e.  ( 1st `  B ) ) )
549, 53sylbid 149 . 2  |-  ( A 
<P  B  ->  ( z  e.  ( 1st `  ( A  +P.  C ) )  ->  z  e.  ( 1st `  B ) ) )
5554ssrdv 3032 1  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  C
) )  C_  ( 1st `  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 3000   <.cop 3453   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900    +Q cplq 6902    <Q cltq 6905   P.cnp 6911    +P. cpp 6913    <P cltp 6915
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 3960  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
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 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-eprel 4125  df-id 4129  df-po 4132  df-iso 4133  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-recs 6084  df-irdg 6149  df-1o 6195  df-2o 6196  df-oadd 6199  df-omul 6200  df-er 6306  df-ec 6308  df-qs 6312  df-ni 6924  df-pli 6925  df-mi 6926  df-lti 6927  df-plpq 6964  df-mpq 6965  df-enq 6967  df-nqqs 6968  df-plqqs 6969  df-mqqs 6970  df-1nqqs 6971  df-rq 6972  df-ltnqqs 6973  df-enq0 7044  df-nq0 7045  df-0nq0 7046  df-plq0 7047  df-mq0 7048  df-inp 7086  df-iplp 7088  df-iltp 7090
This theorem is referenced by:  ltexpri  7233
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