ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltexpri Unicode version

Theorem ltexpri 7575
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 2239 . . . . . . 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 5871 . . . . . . . 8  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  +Q  y
)  =  ( v  +Q  u ) )
54eleq1d 2239 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 1st `  B )  <-> 
( v  +Q  u
)  e.  ( 1st `  B ) ) )
62, 5anbi12d 470 . . . . . 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 1922 . . . . 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 2729 . . . 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 2239 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( z  e.  ( 1st `  A )  <-> 
v  e.  ( 1st `  A ) ) )
104eleq1d 2239 . . . . . . 7  |-  ( ( y  =  u  /\  z  =  v )  ->  ( ( z  +Q  y )  e.  ( 2nd `  B )  <-> 
( v  +Q  u
)  e.  ( 2nd `  B ) ) )
119, 10anbi12d 470 . . . . . 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 1922 . . . . 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 2729 . . . 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 3770 . . 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 7570 . 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 7571 . . . 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 7572 . . . 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 3164 . . 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 7573 . . . 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 7574 . . . 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 3164 . . 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 7467 . . . . . . 7  |-  <P  C_  ( P.  X.  P. )
2322brel 4663 . . . . . 6  |-  ( A 
<P  B  ->  ( A  e.  P.  /\  B  e.  P. ) )
2423simpld 111 . . . . 5  |-  ( A 
<P  B  ->  A  e. 
P. )
25 addclpr 7499 . . . . 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 7434 . . . 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 939 . 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 5861 . . . 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 2179 . . 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 2834 . 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 1348   E.wex 1485    e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242    +Q cplq 7244   P.cnp 7253    +P. cpp 7255    <P cltp 7257
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iplp 7430  df-iltp 7432
This theorem is referenced by:  lteupri  7579  ltaprlem  7580  ltaprg  7581  ltmprr  7604  recexgt0sr  7735  mulgt0sr  7740  map2psrprg  7767
  Copyright terms: Public domain W3C validator