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Theorem ltexprlemfl 6765
Description: Lemma for ltexpri 6769. 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 6661 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4420 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 109 . . . 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 6764 . . . 4  |-  ( A 
<P  B  ->  C  e. 
P. )
6 df-iplp 6624 . . . . 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 6531 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
86, 7genpelvl 6668 . . . 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 397 . . 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 492 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  =  ( w  +Q  u ) )
114ltexprlemell 6754 . . . . . . . . . . 11  |-  ( u  e.  ( 1st `  C
)  <->  ( u  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1211biimpi 117 . . . . . . . . . 10  |-  ( u  e.  ( 1st `  C
)  ->  ( u  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1312ad2antlr 466 . . . . . . . . 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 266 . . . . . . . 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 111 . . . . . . 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 6631 . . . . . . . . . . . . . 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 6643 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
1917, 18syl3an1 1179 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
20193adant2r 1141 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) )  /\  y  e.  ( 2nd `  A ) )  ->  w  <Q  y )
21203adant2r 1141 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
22213adant3r 1143 . . . . . . . . 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 6556 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2423adantl 266 . . . . . . . . . . 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 6521 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
2625brel 4420 . . . . . . . . . . . . 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 109 . . . . . . . . . . 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 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
) ) )  -> 
y  e.  Q. )
30 prop 6631 . . . . . . . . . . . . . . . 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 6637 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3331, 32sylan 271 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3433adantrl 455 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  u  e.  Q. )
3534adantrr 456 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  u  e.  Q. )
36353adant3 935 . . . . . . . . . . 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 6537 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3837adantl 266 . . . . . . . . . . 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 5698 . . . . . . . . . 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 111 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
41 prop 6631 . . . . . . . . . . . . . 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 6640 . . . . . . . . . . . . 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 271 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  u
)  e.  ( 1st `  B ) )  -> 
( ( w  +Q  u )  <Q  (
y  +Q  u )  ->  ( w  +Q  u )  e.  ( 1st `  B ) ) )
4544adantrl 455 . . . . . . . . . . 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 934 . . . . . . . . . 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 143 . . . . . . . . 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 1115 . . . . . . 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 1794 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( w  +Q  u
)  e.  ( 1st `  B ) )
5110, 50eqeltrd 2130 . . . . 5  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  e.  ( 1st `  B ) )
5251expr 361 . . . 4  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  ( z  =  ( w  +Q  u
)  ->  z  e.  ( 1st `  B ) ) )
5352rexlimdvva 2457 . . 3  |-  ( A 
<P  B  ->  ( E. w  e.  ( 1st `  A ) E. u  e.  ( 1st `  C
) z  =  ( w  +Q  u )  ->  z  e.  ( 1st `  B ) ) )
549, 53sylbid 143 . 2  |-  ( A 
<P  B  ->  ( z  e.  ( 1st `  ( A  +P.  C ) )  ->  z  e.  ( 1st `  B ) ) )
5554ssrdv 2979 1  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  C
) )  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   {crab 2327    C_ wss 2945   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    +Q cplq 6438    <Q cltq 6441   P.cnp 6447    +P. cpp 6449    <P cltp 6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-iplp 6624  df-iltp 6626
This theorem is referenced by:  ltexpri  6769
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