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Theorem ltexprlemfl 7610
Description: Lemma for ltexpri 7614. One direction 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 7506 . . . . . 6  |-  <P  C_  ( P.  X.  P. )
21brel 4680 . . . . 5  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
32simpld 112 . . . 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 7609 . . . 4  |-  ( A 
<P  B  ->  C  e. 
P. )
6 df-iplp 7469 . . . . 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 7376 . . . . 5  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g  +Q  h
)  e.  Q. )
86, 7genpelvl 7513 . . . 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 411 . . 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 531 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  =  ( w  +Q  u ) )
114ltexprlemell 7599 . . . . . . . . . . 11  |-  ( u  e.  ( 1st `  C
)  <->  ( u  e. 
Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1211biimpi 120 . . . . . . . . . 10  |-  ( u  e.  ( 1st `  C
)  ->  ( u  e.  Q.  /\  E. y
( y  e.  ( 2nd `  A )  /\  ( y  +Q  u )  e.  ( 1st `  B ) ) ) )
1312ad2antlr 489 . . . . . . . . 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 277 . . . . . . . 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 114 . . . . . . 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 7476 . . . . . . . . . . . . . 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 7488 . . . . . . . . . . . . 13  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
1917, 18syl3an1 1271 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  w  e.  ( 1st `  A )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
20193adant2r 1233 . . . . . . . . . . 11  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) )  /\  y  e.  ( 2nd `  A ) )  ->  w  <Q  y )
21203adant2r 1233 . . . . . . . . . 10  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) )  /\  y  e.  ( 2nd `  A
) )  ->  w  <Q  y )
22213adant3r 1235 . . . . . . . . 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 7401 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
f  <Q  g  <->  ( h  +Q  f )  <Q  (
h  +Q  g ) ) )
2423adantl 277 . . . . . . . . . . 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 7366 . . . . . . . . . . . . . 14  |-  <Q  C_  ( Q.  X.  Q. )
2625brel 4680 . . . . . . . . . . . . 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 112 . . . . . . . . . . 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 114 . . . . . . . . . . 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 7476 . . . . . . . . . . . . . . . 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 7482 . . . . . . . . . . . . . . 15  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3331, 32sylan 283 . . . . . . . . . . . . . 14  |-  ( ( A  <P  B  /\  u  e.  ( 1st `  C ) )  ->  u  e.  Q. )
3433adantrl 478 . . . . . . . . . . . . 13  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  u  e.  Q. )
3534adantrr 479 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  ->  u  e.  Q. )
36353adant3 1017 . . . . . . . . . . 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 7382 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f  +Q  g
)  =  ( g  +Q  f ) )
3837adantl 277 . . . . . . . . . . 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 6046 . . . . . . . . . 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 114 . . . . . . . . . . . . . 14  |-  ( A 
<P  B  ->  B  e. 
P. )
41 prop 7476 . . . . . . . . . . . . . 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 7485 . . . . . . . . . . . . 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 283 . . . . . . . . . . . 12  |-  ( ( A  <P  B  /\  ( y  +Q  u
)  e.  ( 1st `  B ) )  -> 
( ( w  +Q  u )  <Q  (
y  +Q  u )  ->  ( w  +Q  u )  e.  ( 1st `  B ) ) )
4544adantrl 478 . . . . . . . . . . 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 1016 . . . . . . . . . 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 150 . . . . . . . . 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 1203 . . . . . . 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 1898 . . . . . 6  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
( w  +Q  u
)  e.  ( 1st `  B ) )
5110, 50eqeltrd 2254 . . . . 5  |-  ( ( A  <P  B  /\  ( ( w  e.  ( 1st `  A
)  /\  u  e.  ( 1st `  C ) )  /\  z  =  ( w  +Q  u
) ) )  -> 
z  e.  ( 1st `  B ) )
5251expr 375 . . . 4  |-  ( ( A  <P  B  /\  ( w  e.  ( 1st `  A )  /\  u  e.  ( 1st `  C ) ) )  ->  ( z  =  ( w  +Q  u
)  ->  z  e.  ( 1st `  B ) ) )
5352rexlimdvva 2602 . . 3  |-  ( A 
<P  B  ->  ( E. w  e.  ( 1st `  A ) E. u  e.  ( 1st `  C
) z  =  ( w  +Q  u )  ->  z  e.  ( 1st `  B ) ) )
549, 53sylbid 150 . 2  |-  ( A 
<P  B  ->  ( z  e.  ( 1st `  ( A  +P.  C ) )  ->  z  e.  ( 1st `  B ) ) )
5554ssrdv 3163 1  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  C
) )  C_  ( 1st `  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   {crab 2459    C_ wss 3131   <.cop 3597   class class class wbr 4005   ` cfv 5218  (class class class)co 5877   1stc1st 6141   2ndc2nd 6142   Q.cnq 7281    +Q cplq 7283    <Q cltq 7286   P.cnp 7292    +P. cpp 7294    <P cltp 7296
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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-2o 6420  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-enq0 7425  df-nq0 7426  df-0nq0 7427  df-plq0 7428  df-mq0 7429  df-inp 7467  df-iplp 7469  df-iltp 7471
This theorem is referenced by:  ltexpri  7614
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