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Theorem ltexpri 6709
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 103 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  z  =  v )
21eleq1d 2106 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 2nd `  A )  <-> 
v  e.  ( 2nd `  A ) ) )
3 simpl 102 . . . . . . . . 9  |-  ( ( y  =  u  /\  z  =  v )  ->  y  =  u )
41, 3oveq12d 5530 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  +Q  y
)  =  ( v  +Q  u ) )
54eleq1d 2106 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 1st `  B )  <-> 
( v  +Q  u
)  e.  ( 1st `  B ) ) )
62, 5anbi12d 442 . . . . . 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 1804 . . . . 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 2556 . . . 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 2106 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 1st `  A )  <-> 
v  e.  ( 1st `  A ) ) )
104eleq1d 2106 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 2nd `  B )  <-> 
( v  +Q  u
)  e.  ( 2nd `  B ) ) )
119, 10anbi12d 442 . . . . . 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 1804 . . . . 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 2556 . . . 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 3554 . . 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 6704 . 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 6705 . . . 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 6706 . . . 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 2962 . . 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 6707 . . . 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 6708 . . . 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 2962 . . 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 6601 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
2322brel 4392 . . . . . 6  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
2423simpld 105 . . . . 5  |-  ( A 
<P  B  ->  A  e. 
P. )
25 addclpr 6633 . . . . 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 391 . . . 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 107 . . . 4  |-  ( A 
<P  B  ->  B  e. 
P. )
28 preqlu 6568 . . . 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 391 . . 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 851 . 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 5520 . . . 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 2048 . . 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 2656 . 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 391 1  |-  ( A 
<P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   {crab 2310   <.cop 3378   class class class wbr 3764   ` cfv 4902  (class class class)co 5512   1stc1st 5765   2ndc2nd 5766   Q.cnq 6376    +Q cplq 6378   P.cnp 6387    +P. cpp 6389    <P cltp 6391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6400  df-pli 6401  df-mi 6402  df-lti 6403  df-plpq 6440  df-mpq 6441  df-enq 6443  df-nqqs 6444  df-plqqs 6445  df-mqqs 6446  df-1nqqs 6447  df-rq 6448  df-ltnqqs 6449  df-enq0 6520  df-nq0 6521  df-0nq0 6522  df-plq0 6523  df-mq0 6524  df-inp 6562  df-iplp 6564  df-iltp 6566
This theorem is referenced by:  lteupri  6713  ltaprlem  6714  ltaprg  6715  ltmprr  6738  recexgt0sr  6856  mulgt0sr  6860
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