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Theorem ltexpri 7536
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 2226 . . . . . . 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 5845 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  +Q  y
)  =  ( v  +Q  u ) )
54eleq1d 2226 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 1st `  B )  <-> 
( v  +Q  u
)  e.  ( 1st `  B ) ) )
62, 5anbi12d 465 . . . . . 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 1909 . . . . 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 2711 . . . 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 2226 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 1st `  A )  <-> 
v  e.  ( 1st `  A ) ) )
104eleq1d 2226 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 2nd `  B )  <-> 
( v  +Q  u
)  e.  ( 2nd `  B ) ) )
119, 10anbi12d 465 . . . . . 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 1909 . . . . 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 2711 . . . 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 3748 . . 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 7531 . 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 7532 . . . 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 7533 . . . 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 3145 . . 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 7534 . . . 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 7535 . . . 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 3145 . . 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 7428 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
2322brel 4641 . . . . . 6  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
2423simpld 111 . . . . 5  |-  ( A 
<P  B  ->  A  e. 
P. )
25 addclpr 7460 . . . . 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 409 . . . 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 7395 . . . 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 409 . . 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 929 . 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 5835 . . . 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 2166 . . 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 2816 . 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 409 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 1335   E.wex 1472    e. wcel 2128   E.wrex 2436   {crab 2439   <.cop 3564   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   1stc1st 6089   2ndc2nd 6090   Q.cnq 7203    +Q cplq 7205   P.cnp 7214    +P. cpp 7216    <P cltp 7218
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-eprel 4252  df-id 4256  df-po 4259  df-iso 4260  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-1o 6366  df-2o 6367  df-oadd 6370  df-omul 6371  df-er 6483  df-ec 6485  df-qs 6489  df-ni 7227  df-pli 7228  df-mi 7229  df-lti 7230  df-plpq 7267  df-mpq 7268  df-enq 7270  df-nqqs 7271  df-plqqs 7272  df-mqqs 7273  df-1nqqs 7274  df-rq 7275  df-ltnqqs 7276  df-enq0 7347  df-nq0 7348  df-0nq0 7349  df-plq0 7350  df-mq0 7351  df-inp 7389  df-iplp 7391  df-iltp 7393
This theorem is referenced by:  lteupri  7540  ltaprlem  7541  ltaprg  7542  ltmprr  7565  recexgt0sr  7696  mulgt0sr  7701  map2psrprg  7728
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