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Theorem th3qlem2 6761
Description: Lemma for Exercise 44 version of Theorem 3Q of [Enderton] p. 60, extended to operations on ordered pairs. The fourth hypothesis is the compatibility assumption. (Contributed by NM, 4-Aug-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
th3q.1  |-  .~  e.  _V
th3q.2  |-  .~  Er  ( S  X.  S
)
th3q.4  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
Assertion
Ref Expression
th3qlem2  |-  ( ( A  e.  ( ( S  X.  S ) /.  .~  )  /\  B  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
)
Distinct variable groups:    z, w, v, u, t, s, f, g, h,  .~    z, S, w, v, u, t, s, f, g, h   
z, A, w, v, u, t, s, f   
z, B, w, v, u, t, s, f   
z,  .+ , w, v, u, t, s, f, g, h
Allowed substitution hints:    A( g, h)    B( g, h)

Proof of Theorem th3qlem2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 th3q.2 . . 3  |-  .~  Er  ( S  X.  S
)
2 eqid 2284 . . . . 5  |-  ( S  X.  S )  =  ( S  X.  S
)
3 breq1 4027 . . . . . . . 8  |-  ( <.
w ,  v >.  =  s  ->  ( <.
w ,  v >.  .~  <. u ,  t
>. 
<->  s  .~  <. u ,  t >. )
)
43anbi1d 685 . . . . . . 7  |-  ( <.
w ,  v >.  =  s  ->  ( (
<. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  y
)  <->  ( s  .~  <.
u ,  t >.  /\  x  .~  y
) ) )
5 oveq1 5827 . . . . . . . 8  |-  ( <.
w ,  v >.  =  s  ->  ( <.
w ,  v >.  .+  x )  =  ( s  .+  x ) )
65breq1d 4034 . . . . . . 7  |-  ( <.
w ,  v >.  =  s  ->  ( (
<. w ,  v >.  .+  x )  .~  ( <. u ,  t >.  .+  y )  <->  ( s  .+  x )  .~  ( <. u ,  t >.  .+  y ) ) )
74, 6imbi12d 311 . . . . . 6  |-  ( <.
w ,  v >.  =  s  ->  ( ( ( <. w ,  v
>.  .~  <. u ,  t
>.  /\  x  .~  y
)  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) )  <->  ( (
s  .~  <. u ,  t >.  /\  x  .~  y )  ->  (
s  .+  x )  .~  ( <. u ,  t
>.  .+  y ) ) ) )
87imbi2d 307 . . . . 5  |-  ( <.
w ,  v >.  =  s  ->  ( ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  y
)  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) ) )  <->  ( (
x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  -> 
( ( s  .~  <.
u ,  t >.  /\  x  .~  y
)  ->  ( s  .+  x )  .~  ( <. u ,  t >.  .+  y ) ) ) ) )
9 breq2 4028 . . . . . . . 8  |-  ( <.
u ,  t >.  =  f  ->  ( s  .~  <. u ,  t
>. 
<->  s  .~  f ) )
109anbi1d 685 . . . . . . 7  |-  ( <.
u ,  t >.  =  f  ->  ( ( s  .~  <. u ,  t >.  /\  x  .~  y )  <->  ( s  .~  f  /\  x  .~  y ) ) )
11 oveq1 5827 . . . . . . . 8  |-  ( <.
u ,  t >.  =  f  ->  ( <.
u ,  t >.  .+  y )  =  ( f  .+  y ) )
1211breq2d 4036 . . . . . . 7  |-  ( <.
u ,  t >.  =  f  ->  ( ( s  .+  x )  .~  ( <. u ,  t >.  .+  y
)  <->  ( s  .+  x )  .~  (
f  .+  y )
) )
1310, 12imbi12d 311 . . . . . 6  |-  ( <.
u ,  t >.  =  f  ->  ( ( ( s  .~  <. u ,  t >.  /\  x  .~  y )  ->  (
s  .+  x )  .~  ( <. u ,  t
>.  .+  y ) )  <-> 
( ( s  .~  f  /\  x  .~  y
)  ->  ( s  .+  x )  .~  (
f  .+  y )
) ) )
1413imbi2d 307 . . . . 5  |-  ( <.
u ,  t >.  =  f  ->  ( ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  ->  ( (
s  .~  <. u ,  t >.  /\  x  .~  y )  ->  (
s  .+  x )  .~  ( <. u ,  t
>.  .+  y ) ) )  <->  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S
) )  ->  (
( s  .~  f  /\  x  .~  y
)  ->  ( s  .+  x )  .~  (
f  .+  y )
) ) ) )
15 breq1 4027 . . . . . . . . . 10  |-  ( <.
s ,  f >.  =  x  ->  ( <.
s ,  f >.  .~  <. g ,  h >.  <-> 
x  .~  <. g ,  h >. ) )
1615anbi2d 684 . . . . . . . . 9  |-  ( <.
s ,  f >.  =  x  ->  ( (
<. w ,  v >.  .~  <. u ,  t
>.  /\  <. s ,  f
>.  .~  <. g ,  h >. )  <->  ( <. w ,  v >.  .~  <. u ,  t >.  /\  x  .~  <. g ,  h >. ) ) )
17 oveq2 5828 . . . . . . . . . 10  |-  ( <.
s ,  f >.  =  x  ->  ( <.
w ,  v >.  .+  <. s ,  f
>. )  =  ( <. w ,  v >.  .+  x ) )
1817breq1d 4034 . . . . . . . . 9  |-  ( <.
s ,  f >.  =  x  ->  ( (
<. w ,  v >.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. )  <->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  <. g ,  h >. )
) )
1916, 18imbi12d 311 . . . . . . . 8  |-  ( <.
s ,  f >.  =  x  ->  ( ( ( <. w ,  v
>.  .~  <. u ,  t
>.  /\  <. s ,  f
>.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+  <. s ,  f >. )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) )  <->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  x )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) ) ) )
2019imbi2d 307 . . . . . . 7  |-  ( <.
s ,  f >.  =  x  ->  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  <. s ,  f
>.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+  <. s ,  f >. )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) ) )  <->  ( (
( w  e.  S  /\  v  e.  S
)  /\  ( u  e.  S  /\  t  e.  S ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  x  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  <. g ,  h >. )
) ) ) )
21 breq2 4028 . . . . . . . . . 10  |-  ( <.
g ,  h >.  =  y  ->  ( x  .~  <. g ,  h >.  <-> 
x  .~  y )
)
2221anbi2d 684 . . . . . . . . 9  |-  ( <.
g ,  h >.  =  y  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  <. g ,  h >. )  <->  (
<. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  y
) ) )
23 oveq2 5828 . . . . . . . . . 10  |-  ( <.
g ,  h >.  =  y  ->  ( <. u ,  t >.  .+  <. g ,  h >. )  =  ( <. u ,  t >.  .+  y
) )
2423breq2d 4036 . . . . . . . . 9  |-  ( <.
g ,  h >.  =  y  ->  ( ( <. w ,  v >.  .+  x )  .~  ( <. u ,  t >.  .+  <. g ,  h >. )  <->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) ) )
2522, 24imbi12d 311 . . . . . . . 8  |-  ( <.
g ,  h >.  =  y  ->  ( (
( <. w ,  v
>.  .~  <. u ,  t
>.  /\  x  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  x )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) )  <->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  y
)  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) ) ) )
2625imbi2d 307 . . . . . . 7  |-  ( <.
g ,  h >.  =  y  ->  ( (
( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  x )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) ) )  <->  ( (
( w  e.  S  /\  v  e.  S
)  /\  ( u  e.  S  /\  t  e.  S ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  x  .~  y )  ->  ( <. w ,  v >.  .+  x )  .~  ( <. u ,  t >.  .+  y ) ) ) ) )
27 th3q.4 . . . . . . . 8  |-  ( ( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  /\  ( (
s  e.  S  /\  f  e.  S )  /\  ( g  e.  S  /\  h  e.  S
) ) )  -> 
( ( <. w ,  v >.  .~  <. u ,  t >.  /\  <. s ,  f >.  .~  <. g ,  h >. )  ->  ( <. w ,  v
>.  .+  <. s ,  f
>. )  .~  ( <. u ,  t >.  .+  <. g ,  h >. ) ) )
2827expcom 424 . . . . . . 7  |-  ( ( ( s  e.  S  /\  f  e.  S
)  /\  ( g  e.  S  /\  h  e.  S ) )  -> 
( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  <. s ,  f
>.  .~  <. g ,  h >. )  ->  ( <. w ,  v >.  .+  <. s ,  f >. )  .~  ( <. u ,  t
>.  .+  <. g ,  h >. ) ) ) )
292, 20, 26, 282optocl 4764 . . . . . 6  |-  ( ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S ) )  -> 
( ( ( w  e.  S  /\  v  e.  S )  /\  (
u  e.  S  /\  t  e.  S )
)  ->  ( ( <. w ,  v >.  .~  <. u ,  t
>.  /\  x  .~  y
)  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) ) ) )
3029com12 27 . . . . 5  |-  ( ( ( w  e.  S  /\  v  e.  S
)  /\  ( u  e.  S  /\  t  e.  S ) )  -> 
( ( x  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  ->  (
( <. w ,  v
>.  .~  <. u ,  t
>.  /\  x  .~  y
)  ->  ( <. w ,  v >.  .+  x
)  .~  ( <. u ,  t >.  .+  y
) ) ) )
312, 8, 14, 302optocl 4764 . . . 4  |-  ( ( s  e.  ( S  X.  S )  /\  f  e.  ( S  X.  S ) )  -> 
( ( x  e.  ( S  X.  S
)  /\  y  e.  ( S  X.  S
) )  ->  (
( s  .~  f  /\  x  .~  y
)  ->  ( s  .+  x )  .~  (
f  .+  y )
) ) )
3231imp 418 . . 3  |-  ( ( ( s  e.  ( S  X.  S )  /\  f  e.  ( S  X.  S ) )  /\  ( x  e.  ( S  X.  S )  /\  y  e.  ( S  X.  S
) ) )  -> 
( ( s  .~  f  /\  x  .~  y
)  ->  ( s  .+  x )  .~  (
f  .+  y )
) )
331, 32th3qlem1 6760 . 2  |-  ( ( A  e.  ( ( S  X.  S ) /.  .~  )  /\  B  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. s E. x
( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [ ( s  .+  x ) ]  .~  ) )
34 opex 4236 . . . . . 6  |-  <. w ,  v >.  e.  _V
35 opex 4236 . . . . . 6  |-  <. u ,  t >.  e.  _V
36 eceq1 6692 . . . . . . . . 9  |-  ( s  =  <. w ,  v
>.  ->  [ s ]  .~  =  [ <. w ,  v >. ]  .~  )
3736eqeq2d 2295 . . . . . . . 8  |-  ( s  =  <. w ,  v
>.  ->  ( A  =  [ s ]  .~  <->  A  =  [ <. w ,  v >. ]  .~  ) )
38 eceq1 6692 . . . . . . . . 9  |-  ( x  =  <. u ,  t
>.  ->  [ x ]  .~  =  [ <. u ,  t >. ]  .~  )
3938eqeq2d 2295 . . . . . . . 8  |-  ( x  =  <. u ,  t
>.  ->  ( B  =  [ x ]  .~  <->  B  =  [ <. u ,  t >. ]  .~  ) )
4037, 39bi2anan9 843 . . . . . . 7  |-  ( ( s  =  <. w ,  v >.  /\  x  =  <. u ,  t
>. )  ->  ( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  ) 
<->  ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  ) ) )
41 oveq12 5829 . . . . . . . . 9  |-  ( ( s  =  <. w ,  v >.  /\  x  =  <. u ,  t
>. )  ->  ( s 
.+  x )  =  ( <. w ,  v
>.  .+  <. u ,  t
>. ) )
42 eceq1 6692 . . . . . . . . 9  |-  ( ( s  .+  x )  =  ( <. w ,  v >.  .+  <. u ,  t >. )  ->  [ ( s  .+  x ) ]  .~  =  [ ( <. w ,  v >.  .+  <. u ,  t >. ) ]  .~  )
4341, 42syl 15 . . . . . . . 8  |-  ( ( s  =  <. w ,  v >.  /\  x  =  <. u ,  t
>. )  ->  [ ( s  .+  x ) ]  .~  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
4443eqeq2d 2295 . . . . . . 7  |-  ( ( s  =  <. w ,  v >.  /\  x  =  <. u ,  t
>. )  ->  ( z  =  [ ( s 
.+  x ) ]  .~  <->  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
4540, 44anbi12d 691 . . . . . 6  |-  ( ( s  =  <. w ,  v >.  /\  x  =  <. u ,  t
>. )  ->  ( ( ( A  =  [
s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [ ( s  .+  x ) ]  .~  ) 
<->  ( ( A  =  [ <. w ,  v
>. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
) )
4634, 35, 45spc2ev 2877 . . . . 5  |-  ( ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )  ->  E. s E. x
( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [ ( s  .+  x ) ]  .~  ) )
4746exlimivv 1668 . . . 4  |-  ( E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )  ->  E. s E. x
( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [ ( s  .+  x ) ]  .~  ) )
4847exlimivv 1668 . . 3  |-  ( E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )  ->  E. s E. x
( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [ ( s  .+  x ) ]  .~  ) )
4948moimi 2191 . 2  |-  ( E* z E. s E. x ( ( A  =  [ s ]  .~  /\  B  =  [ x ]  .~  )  /\  z  =  [
( s  .+  x
) ]  .~  )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t >. ]  .~  )  /\  z  =  [
( <. w ,  v
>.  .+  <. u ,  t
>. ) ]  .~  )
)
5033, 49syl 15 1  |-  ( ( A  e.  ( ( S  X.  S ) /.  .~  )  /\  B  e.  ( ( S  X.  S ) /.  .~  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ]  .~  /\  B  =  [ <. u ,  t
>. ]  .~  )  /\  z  =  [ ( <. w ,  v >.  .+  <. u ,  t
>. ) ]  .~  )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   E*wmo 2145   _Vcvv 2789   <.cop 3644   class class class wbr 4024    X. cxp 4686  (class class class)co 5820    Er wer 6653   [cec 6654   /.cqs 6655
This theorem is referenced by:  th3qcor  6762  th3q  6763
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fv 5229  df-ov 5823  df-er 6656  df-ec 6658  df-qs 6662
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