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Theorem ltexpri 7385
Description: Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.)
Assertion
Ref Expression
ltexpri  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltexpri
Dummy variables  y  z  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 109 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  z  =  v )
21eleq1d 2184 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 2nd `  A )  <-> 
v  e.  ( 2nd `  A ) ) )
3 simpl 108 . . . . . . . . 9  |-  ( ( y  =  u  /\  z  =  v )  ->  y  =  u )
41, 3oveq12d 5758 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  +Q  y
)  =  ( v  +Q  u ) )
54eleq1d 2184 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 1st `  B )  <-> 
( v  +Q  u
)  e.  ( 1st `  B ) ) )
62, 5anbi12d 462 . . . . . 6  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) )  <->  ( v  e.  ( 2nd `  A
)  /\  ( v  +Q  u )  e.  ( 1st `  B ) ) ) )
76cbvexdva 1879 . . . . 5  |-  ( y  =  u  ->  ( E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) )  <->  E. v ( v  e.  ( 2nd `  A
)  /\  ( v  +Q  u )  e.  ( 1st `  B ) ) ) )
87cbvrabv 2657 . . . 4  |-  { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) }  =  {
u  e.  Q.  |  E. v ( v  e.  ( 2nd `  A
)  /\  ( v  +Q  u )  e.  ( 1st `  B ) ) }
91eleq1d 2184 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 1st `  A )  <-> 
v  e.  ( 1st `  A ) ) )
104eleq1d 2184 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 2nd `  B )  <-> 
( v  +Q  u
)  e.  ( 2nd `  B ) ) )
119, 10anbi12d 462 . . . . . 6  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) )  <->  ( v  e.  ( 1st `  A
)  /\  ( v  +Q  u )  e.  ( 2nd `  B ) ) ) )
1211cbvexdva 1879 . . . . 5  |-  ( y  =  u  ->  ( E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) )  <->  E. v ( v  e.  ( 1st `  A
)  /\  ( v  +Q  u )  e.  ( 2nd `  B ) ) ) )
1312cbvrabv 2657 . . . 4  |-  { y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) }  =  {
u  e.  Q.  |  E. v ( v  e.  ( 1st `  A
)  /\  ( v  +Q  u )  e.  ( 2nd `  B ) ) }
148, 13opeq12i 3678 . . 3  |-  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  =  <. { u  e.  Q.  |  E. v ( v  e.  ( 2nd `  A
)  /\  ( v  +Q  u )  e.  ( 1st `  B ) ) } ,  {
u  e.  Q.  |  E. v ( v  e.  ( 1st `  A
)  /\  ( v  +Q  u )  e.  ( 2nd `  B ) ) } >.
1514ltexprlempr 7380 . 2  |-  ( A 
<P  B  ->  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  e.  P. )
1614ltexprlemfl 7381 . . . 4  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  C_  ( 1st `  B ) )
1714ltexprlemrl 7382 . . . 4  |-  ( A 
<P  B  ->  ( 1st `  B )  C_  ( 1st `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
) )
1816, 17eqssd 3082 . . 3  |-  ( A 
<P  B  ->  ( 1st `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 1st `  B ) )
1914ltexprlemfu 7383 . . . 4  |-  ( A 
<P  B  ->  ( 2nd `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  C_  ( 2nd `  B ) )
2014ltexprlemru 7384 . . . 4  |-  ( A 
<P  B  ->  ( 2nd `  B )  C_  ( 2nd `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
) )
2119, 20eqssd 3082 . . 3  |-  ( A 
<P  B  ->  ( 2nd `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 2nd `  B ) )
22 ltrelpr 7277 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
2322brel 4559 . . . . . 6  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
2423simpld 111 . . . . 5  |-  ( A 
<P  B  ->  A  e. 
P. )
25 addclpr 7309 . . . . 5  |-  ( ( A  e.  P.  /\  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  e.  P. )  ->  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  e.  P. )
2624, 15, 25syl2anc 406 . . . 4  |-  ( A 
<P  B  ->  ( A  +P.  <. { y  e. 
Q.  |  E. z
( z  e.  ( 2nd `  A )  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  e.  P. )
2723simprd 113 . . . 4  |-  ( A 
<P  B  ->  B  e. 
P. )
28 preqlu 7244 . . . 4  |-  ( ( ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  e.  P.  /\  B  e. 
P. )  ->  (
( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  =  B  <->  ( ( 1st `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 1st `  B )  /\  ( 2nd `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 2nd `  B ) ) ) )
2926, 27, 28syl2anc 406 . . 3  |-  ( A 
<P  B  ->  ( ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  =  B  <->  ( ( 1st `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 1st `  B )  /\  ( 2nd `  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)  =  ( 2nd `  B ) ) ) )
3018, 21, 29mpbir2and 911 . 2  |-  ( A 
<P  B  ->  ( A  +P.  <. { y  e. 
Q.  |  E. z
( z  e.  ( 2nd `  A )  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  =  B )
31 oveq2 5748 . . . 4  |-  ( x  =  <. { y  e. 
Q.  |  E. z
( z  e.  ( 2nd `  A )  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  ->  ( A  +P.  x )  =  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )
)
3231eqeq1d 2124 . . 3  |-  ( x  =  <. { y  e. 
Q.  |  E. z
( z  e.  ( 2nd `  A )  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  ->  (
( A  +P.  x
)  =  B  <->  ( A  +P.  <. { y  e. 
Q.  |  E. z
( z  e.  ( 2nd `  A )  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  =  B ) )
3332rspcev 2761 . 2  |-  ( (
<. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >.  e.  P.  /\  ( A  +P.  <. { y  e.  Q.  |  E. z ( z  e.  ( 2nd `  A
)  /\  ( z  +Q  y )  e.  ( 1st `  B ) ) } ,  {
y  e.  Q.  |  E. z ( z  e.  ( 1st `  A
)  /\  ( z  +Q  y )  e.  ( 2nd `  B ) ) } >. )  =  B )  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
3415, 30, 33syl2anc 406 1  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314   E.wex 1451    e. wcel 1463   E.wrex 2392   {crab 2395   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    +Q cplq 7054   P.cnp 7063    +P. cpp 7065    <P cltp 7067
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-2o 6280  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-pli 7077  df-mi 7078  df-lti 7079  df-plpq 7116  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-plqqs 7121  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-enq0 7196  df-nq0 7197  df-0nq0 7198  df-plq0 7199  df-mq0 7200  df-inp 7238  df-iplp 7240  df-iltp 7242
This theorem is referenced by:  lteupri  7389  ltaprlem  7390  ltaprg  7391  ltmprr  7414  recexgt0sr  7545  mulgt0sr  7550  map2psrprg  7577
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