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Theorem eroveu 6488
Description: Lemma for eroprf 6490. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
eropr.1  |-  J  =  ( A /. R
)
eropr.2  |-  K  =  ( B /. S
)
eropr.3  |-  ( ph  ->  T  e.  Z )
eropr.4  |-  ( ph  ->  R  Er  U )
eropr.5  |-  ( ph  ->  S  Er  V )
eropr.6  |-  ( ph  ->  T  Er  W )
eropr.7  |-  ( ph  ->  A  C_  U )
eropr.8  |-  ( ph  ->  B  C_  V )
eropr.9  |-  ( ph  ->  C  C_  W )
eropr.10  |-  ( ph  ->  .+  : ( A  X.  B ) --> C )
eropr.11  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r R s  /\  t S u )  ->  (
r  .+  t ) T ( s  .+  u ) ) )
Assertion
Ref Expression
eroveu  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
Distinct variable groups:    q, p, r, s, t, u, z, A    B, p, q, r, s, t, u, z    J, p, q, z    R, p, q, r, s, t, u, z    K, p, q, z    S, p, q, r, s, t, u, z    .+ , p, q, r, s, t, u, z    ph, p, q, r, s, t, u, z    T, p, q, r, s, t, u, z    X, p, q, r, s, t, u, z    Y, p, q, r, s, t, u, z
Allowed substitution hints:    C( z, u, t, s, r, q, p)    U( z, u, t, s, r, q, p)    J( u, t, s, r)    K( u, t, s, r)    V( z, u, t, s, r, q, p)    W( z, u, t, s, r, q, p)    Z( z, u, t, s, r, q, p)

Proof of Theorem eroveu
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elqsi 6449 . . . . . . . 8  |-  ( X  e.  ( A /. R )  ->  E. p  e.  A  X  =  [ p ] R
)
2 eropr.1 . . . . . . . 8  |-  J  =  ( A /. R
)
31, 2eleq2s 2212 . . . . . . 7  |-  ( X  e.  J  ->  E. p  e.  A  X  =  [ p ] R
)
4 elqsi 6449 . . . . . . . 8  |-  ( Y  e.  ( B /. S )  ->  E. q  e.  B  Y  =  [ q ] S
)
5 eropr.2 . . . . . . . 8  |-  K  =  ( B /. S
)
64, 5eleq2s 2212 . . . . . . 7  |-  ( Y  e.  K  ->  E. q  e.  B  Y  =  [ q ] S
)
73, 6anim12i 336 . . . . . 6  |-  ( ( X  e.  J  /\  Y  e.  K )  ->  ( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
87adantl 275 . . . . 5  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
9 reeanv 2577 . . . . 5  |-  ( E. p  e.  A  E. q  e.  B  ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  <->  ( E. p  e.  A  X  =  [ p ] R  /\  E. q  e.  B  Y  =  [ q ] S ) )
108, 9sylibr 133 . . . 4  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E. p  e.  A  E. q  e.  B  ( X  =  [
p ] R  /\  Y  =  [ q ] S ) )
11 eropr.3 . . . . . . . 8  |-  ( ph  ->  T  e.  Z )
1211adantr 274 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  T  e.  Z )
13 ecexg 6401 . . . . . . 7  |-  ( T  e.  Z  ->  [ ( p  .+  q ) ] T  e.  _V )
14 elisset 2674 . . . . . . 7  |-  ( [ ( p  .+  q
) ] T  e. 
_V  ->  E. z  z  =  [ ( p  .+  q ) ] T
)
1512, 13, 143syl 17 . . . . . 6  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E. z  z  =  [ ( p  .+  q ) ] T
)
1615biantrud 302 . . . . 5  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  <-> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  E. z  z  =  [ ( p 
.+  q ) ] T ) ) )
17162rexbidv 2437 . . . 4  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( E. p  e.  A  E. q  e.  B  ( X  =  [ p ] R  /\  Y  =  [
q ] S )  <->  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  E. z  z  =  [ ( p 
.+  q ) ] T ) ) )
1810, 17mpbid 146 . . 3  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  E. z  z  =  [ ( p 
.+  q ) ] T ) )
19 19.42v 1862 . . . . . . . 8  |-  ( E. z ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  E. z  z  =  [ ( p 
.+  q ) ] T ) )
2019bicomi 131 . . . . . . 7  |-  ( ( ( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  E. z  z  =  [
( p  .+  q
) ] T )  <->  E. z ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
2120rexbii 2419 . . . . . 6  |-  ( E. q  e.  B  ( ( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  E. z  z  =  [
( p  .+  q
) ] T )  <->  E. q  e.  B  E. z ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
22 rexcom4 2683 . . . . . 6  |-  ( E. q  e.  B  E. z ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <->  E. z E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
2321, 22bitri 183 . . . . 5  |-  ( E. q  e.  B  ( ( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  E. z  z  =  [
( p  .+  q
) ] T )  <->  E. z E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
2423rexbii 2419 . . . 4  |-  ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  E. z  z  =  [
( p  .+  q
) ] T )  <->  E. p  e.  A  E. z E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
25 rexcom4 2683 . . . 4  |-  ( E. p  e.  A  E. z E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <->  E. z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
2624, 25bitri 183 . . 3  |-  ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  E. z  z  =  [
( p  .+  q
) ] T )  <->  E. z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
2718, 26sylib 121 . 2  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E. z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
28 reeanv 2577 . . . . . 6  |-  ( E. r  e.  A  E. s  e.  A  ( E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T )  /\  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  <->  ( E. r  e.  A  E. t  e.  B  (
( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  /\  E. s  e.  A  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) ) )
29 eceq1 6432 . . . . . . . . . . 11  |-  ( p  =  r  ->  [ p ] R  =  [
r ] R )
3029eqeq2d 2129 . . . . . . . . . 10  |-  ( p  =  r  ->  ( X  =  [ p ] R  <->  X  =  [
r ] R ) )
3130anbi1d 460 . . . . . . . . 9  |-  ( p  =  r  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
q ] S ) ) )
32 oveq1 5749 . . . . . . . . . . 11  |-  ( p  =  r  ->  (
p  .+  q )  =  ( r  .+  q ) )
3332eceq1d 6433 . . . . . . . . . 10  |-  ( p  =  r  ->  [ ( p  .+  q ) ] T  =  [
( r  .+  q
) ] T )
3433eqeq2d 2129 . . . . . . . . 9  |-  ( p  =  r  ->  (
z  =  [ ( p  .+  q ) ] T  <->  z  =  [ ( r  .+  q ) ] T
) )
3531, 34anbi12d 464 . . . . . . . 8  |-  ( p  =  r  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ r ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( r  .+  q
) ] T ) ) )
36 eceq1 6432 . . . . . . . . . . 11  |-  ( q  =  t  ->  [ q ] S  =  [
t ] S )
3736eqeq2d 2129 . . . . . . . . . 10  |-  ( q  =  t  ->  ( Y  =  [ q ] S  <->  Y  =  [
t ] S ) )
3837anbi2d 459 . . . . . . . . 9  |-  ( q  =  t  ->  (
( X  =  [
r ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
t ] S ) ) )
39 oveq2 5750 . . . . . . . . . . 11  |-  ( q  =  t  ->  (
r  .+  q )  =  ( r  .+  t ) )
4039eceq1d 6433 . . . . . . . . . 10  |-  ( q  =  t  ->  [ ( r  .+  q ) ] T  =  [
( r  .+  t
) ] T )
4140eqeq2d 2129 . . . . . . . . 9  |-  ( q  =  t  ->  (
z  =  [ ( r  .+  q ) ] T  <->  z  =  [ ( r  .+  t ) ] T
) )
4238, 41anbi12d 464 . . . . . . . 8  |-  ( q  =  t  ->  (
( ( X  =  [ r ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( r  .+  q
) ] T )  <-> 
( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T ) ) )
4335, 42cbvrex2v 2640 . . . . . . 7  |-  ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  <->  E. r  e.  A  E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T ) )
44 eceq1 6432 . . . . . . . . . . 11  |-  ( p  =  s  ->  [ p ] R  =  [
s ] R )
4544eqeq2d 2129 . . . . . . . . . 10  |-  ( p  =  s  ->  ( X  =  [ p ] R  <->  X  =  [
s ] R ) )
4645anbi1d 460 . . . . . . . . 9  |-  ( p  =  s  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
q ] S ) ) )
47 oveq1 5749 . . . . . . . . . . 11  |-  ( p  =  s  ->  (
p  .+  q )  =  ( s  .+  q ) )
4847eceq1d 6433 . . . . . . . . . 10  |-  ( p  =  s  ->  [ ( p  .+  q ) ] T  =  [
( s  .+  q
) ] T )
4948eqeq2d 2129 . . . . . . . . 9  |-  ( p  =  s  ->  (
w  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( s  .+  q ) ] T
) )
5046, 49anbi12d 464 . . . . . . . 8  |-  ( p  =  s  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ s ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( s  .+  q
) ] T ) ) )
51 eceq1 6432 . . . . . . . . . . 11  |-  ( q  =  u  ->  [ q ] S  =  [
u ] S )
5251eqeq2d 2129 . . . . . . . . . 10  |-  ( q  =  u  ->  ( Y  =  [ q ] S  <->  Y  =  [
u ] S ) )
5352anbi2d 459 . . . . . . . . 9  |-  ( q  =  u  ->  (
( X  =  [
s ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
u ] S ) ) )
54 oveq2 5750 . . . . . . . . . . 11  |-  ( q  =  u  ->  (
s  .+  q )  =  ( s  .+  u ) )
5554eceq1d 6433 . . . . . . . . . 10  |-  ( q  =  u  ->  [ ( s  .+  q ) ] T  =  [
( s  .+  u
) ] T )
5655eqeq2d 2129 . . . . . . . . 9  |-  ( q  =  u  ->  (
w  =  [ ( s  .+  q ) ] T  <->  w  =  [ ( s  .+  u ) ] T
) )
5753, 56anbi12d 464 . . . . . . . 8  |-  ( q  =  u  ->  (
( ( X  =  [ s ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( s  .+  q
) ] T )  <-> 
( ( X  =  [ s ] R  /\  Y  =  [
u ] S )  /\  w  =  [
( s  .+  u
) ] T ) ) )
5850, 57cbvrex2v 2640 . . . . . . 7  |-  ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  w  =  [ ( p  .+  q ) ] T
)  <->  E. s  e.  A  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [
u ] S )  /\  w  =  [
( s  .+  u
) ] T ) )
5943, 58anbi12i 455 . . . . . 6  |-  ( ( E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) )  <->  ( E. r  e.  A  E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  E. s  e.  A  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) ) )
6028, 59bitr4i 186 . . . . 5  |-  ( E. r  e.  A  E. s  e.  A  ( E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T )  /\  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  <->  ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  w  =  [ ( p  .+  q ) ] T
) ) )
61 reeanv 2577 . . . . . . 7  |-  ( E. t  e.  B  E. u  e.  B  (
( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T )  /\  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  <->  ( E. t  e.  B  (
( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  /\  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) ) )
62 eropr.11 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r R s  /\  t S u )  ->  (
r  .+  t ) T ( s  .+  u ) ) )
63 eropr.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  Er  U )
6463adantr 274 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  R  Er  U )
65 eropr.7 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A  C_  U )
6665adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  A  C_  U )
67 simprll 511 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  A )
6866, 67sseldd 3068 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  U )
6964, 68erth 6441 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r R s  <->  [ r ] R  =  [ s ] R
) )
70 eropr.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  Er  V )
7170adantr 274 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  S  Er  V )
72 eropr.8 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  B  C_  V )
7372adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  B  C_  V )
74 simprrl 513 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  B )
7573, 74sseldd 3068 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  V )
7671, 75erth 6441 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( t S u  <->  [ t ] S  =  [ u ] S
) )
7769, 76anbi12d 464 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r R s  /\  t S u )  <->  ( [
r ] R  =  [ s ] R  /\  [ t ] S  =  [ u ] S
) ) )
78 eropr.6 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  T  Er  W )
7978adantr 274 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  T  Er  W )
80 eropr.9 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  C  C_  W )
8180adantr 274 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  C  C_  W )
82 eropr.10 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  .+  : ( A  X.  B ) --> C )
8382adantr 274 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  .+  : ( A  X.  B ) --> C )
8483, 67, 74fovrnd 5883 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  C )
8581, 84sseldd 3068 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  W )
8679, 85erth 6441 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r  .+  t ) T ( s  .+  u )  <->  [ ( r  .+  t ) ] T  =  [ ( s  .+  u ) ] T
) )
8762, 77, 863imtr3d 201 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( [ r ] R  =  [
s ] R  /\  [ t ] S  =  [ u ] S
)  ->  [ (
r  .+  t ) ] T  =  [
( s  .+  u
) ] T ) )
88 eqeq2 2127 . . . . . . . . . . . . . 14  |-  ( w  =  [ ( s 
.+  u ) ] T  ->  ( [
( r  .+  t
) ] T  =  w  <->  [ ( r  .+  t ) ] T  =  [ ( s  .+  u ) ] T
) )
8988biimprcd 159 . . . . . . . . . . . . 13  |-  ( [ ( r  .+  t
) ] T  =  [ ( s  .+  u ) ] T  ->  ( w  =  [
( s  .+  u
) ] T  ->  [ ( r  .+  t ) ] T  =  w ) )
9087, 89syl6 33 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( [ r ] R  =  [
s ] R  /\  [ t ] S  =  [ u ] S
)  ->  ( w  =  [ ( s  .+  u ) ] T  ->  [ ( r  .+  t ) ] T  =  w ) ) )
9190impd 252 . . . . . . . . . . 11  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( ( [ r ] R  =  [ s ] R  /\  [ t ] S  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
)  ->  [ (
r  .+  t ) ] T  =  w
) )
92 eqeq1 2124 . . . . . . . . . . . . . . 15  |-  ( X  =  [ r ] R  ->  ( X  =  [ s ] R  <->  [ r ] R  =  [ s ] R
) )
93 eqeq1 2124 . . . . . . . . . . . . . . 15  |-  ( Y  =  [ t ] S  ->  ( Y  =  [ u ] S  <->  [ t ] S  =  [ u ] S
) )
9492, 93bi2anan9 580 . . . . . . . . . . . . . 14  |-  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  ->  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  <->  ( [ r ] R  =  [
s ] R  /\  [ t ] S  =  [ u ] S
) ) )
9594anbi1d 460 . . . . . . . . . . . . 13  |-  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  ->  ( (
( X  =  [
s ] R  /\  Y  =  [ u ] S )  /\  w  =  [ ( s  .+  u ) ] T
)  <->  ( ( [ r ] R  =  [ s ] R  /\  [ t ] S  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) ) )
9695adantr 274 . . . . . . . . . . . 12  |-  ( ( ( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( (
( X  =  [
s ] R  /\  Y  =  [ u ] S )  /\  w  =  [ ( s  .+  u ) ] T
)  <->  ( ( [ r ] R  =  [ s ] R  /\  [ t ] S  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) ) )
97 eqeq1 2124 . . . . . . . . . . . . 13  |-  ( z  =  [ ( r 
.+  t ) ] T  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
9897adantl 275 . . . . . . . . . . . 12  |-  ( ( ( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
9996, 98imbi12d 233 . . . . . . . . . . 11  |-  ( ( ( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( (
( ( X  =  [ s ] R  /\  Y  =  [
u ] S )  /\  w  =  [
( s  .+  u
) ] T )  ->  z  =  w )  <->  ( ( ( [ r ] R  =  [ s ] R  /\  [ t ] S  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
)  ->  [ (
r  .+  t ) ] T  =  w
) ) )
10091, 99syl5ibrcom 156 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( (
( X  =  [
s ] R  /\  Y  =  [ u ] S )  /\  w  =  [ ( s  .+  u ) ] T
)  ->  z  =  w ) ) )
101100impd 252 . . . . . . . . 9  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  ->  z  =  w ) )
102101anassrs 397 . . . . . . . 8  |-  ( ( ( ph  /\  (
r  e.  A  /\  s  e.  A )
)  /\  ( t  e.  B  /\  u  e.  B ) )  -> 
( ( ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  ->  z  =  w ) )
103102rexlimdvva 2534 . . . . . . 7  |-  ( (
ph  /\  ( r  e.  A  /\  s  e.  A ) )  -> 
( E. t  e.  B  E. u  e.  B  ( ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  ->  z  =  w ) )
10461, 103syl5bir 152 . . . . . 6  |-  ( (
ph  /\  ( r  e.  A  /\  s  e.  A ) )  -> 
( ( E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  ->  z  =  w ) )
105104rexlimdvva 2534 . . . . 5  |-  ( ph  ->  ( E. r  e.  A  E. s  e.  A  ( E. t  e.  B  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  /\  z  =  [ ( r  .+  t ) ] T
)  /\  E. u  e.  B  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  /\  w  =  [ ( s  .+  u ) ] T
) )  ->  z  =  w ) )
10660, 105syl5bir 152 . . . 4  |-  ( ph  ->  ( ( E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
)  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  w  =  [ ( p  .+  q ) ] T
) )  ->  z  =  w ) )
107106adantr 274 . . 3  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( ( E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
)  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  w  =  [ ( p  .+  q ) ] T
) )  ->  z  =  w ) )
108107alrimivv 1831 . 2  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  A. z A. w ( ( E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) )  ->  z  =  w ) )
109 eqeq1 2124 . . . . 5  |-  ( z  =  w  ->  (
z  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( p  .+  q ) ] T
) )
110109anbi2d 459 . . . 4  |-  ( z  =  w  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) ) )
1111102rexbidv 2437 . . 3  |-  ( z  =  w  ->  ( E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <->  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) ) )
112111eu4 2039 . 2  |-  ( E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( E. z E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  /\  A. z A. w ( ( E. p  e.  A  E. q  e.  B  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  /\  E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [ q ] S
)  /\  w  =  [ ( p  .+  q ) ] T
) )  ->  z  =  w ) ) )
11327, 108, 112sylanbrc 413 1  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316   E.wex 1453    e. wcel 1465   E!weu 1977   E.wrex 2394   _Vcvv 2660    C_ wss 3041   class class class wbr 3899    X. cxp 4507   -->wf 5089  (class class class)co 5742    Er wer 6394   [cec 6395   /.cqs 6396
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-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-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  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-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  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-fv 5101  df-ov 5745  df-er 6397  df-ec 6399  df-qs 6403
This theorem is referenced by:  erovlem  6489  eroprf  6490
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