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Theorem ltexpri 7389
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 2186 . . . . . . 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 5760 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  +Q  y
)  =  ( v  +Q  u ) )
54eleq1d 2186 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 1st `  B )  <-> 
( v  +Q  u
)  e.  ( 1st `  B ) ) )
62, 5anbi12d 464 . . . . . 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 1881 . . . . 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 2659 . . . 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 2186 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 1st `  A )  <-> 
v  e.  ( 1st `  A ) ) )
104eleq1d 2186 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 2nd `  B )  <-> 
( v  +Q  u
)  e.  ( 2nd `  B ) ) )
119, 10anbi12d 464 . . . . . 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 1881 . . . . 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 2659 . . . 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 3680 . . 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 7384 . 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 7385 . . . 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 7386 . . . 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 3084 . . 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 7387 . . . 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 7388 . . . 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 3084 . . 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 7281 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
2322brel 4561 . . . . . 6  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
2423simpld 111 . . . . 5  |-  ( A 
<P  B  ->  A  e. 
P. )
25 addclpr 7313 . . . . 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 408 . . . 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 7248 . . . 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 408 . . 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 913 . 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 5750 . . . 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 2126 . . 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 2763 . 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 408 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 1316   E.wex 1453    e. wcel 1465   E.wrex 2394   {crab 2397   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056    +Q cplq 7058   P.cnp 7067    +P. cpp 7069    <P cltp 7071
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-2o 6282  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-pli 7081  df-mi 7082  df-lti 7083  df-plpq 7120  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-plqqs 7125  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-enq0 7200  df-nq0 7201  df-0nq0 7202  df-plq0 7203  df-mq0 7204  df-inp 7242  df-iplp 7244  df-iltp 7246
This theorem is referenced by:  lteupri  7393  ltaprlem  7394  ltaprg  7395  ltmprr  7418  recexgt0sr  7549  mulgt0sr  7554  map2psrprg  7581
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