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Theorem eroveu 6335
Description: Lemma for eroprf 6337. (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 6296 . . . . . . . 8  |-  ( X  e.  ( A /. R )  ->  E. p  e.  A  X  =  [ p ] R
)
2 eropr.1 . . . . . . . 8  |-  J  =  ( A /. R
)
31, 2eleq2s 2179 . . . . . . 7  |-  ( X  e.  J  ->  E. p  e.  A  X  =  [ p ] R
)
4 elqsi 6296 . . . . . . . 8  |-  ( Y  e.  ( B /. S )  ->  E. q  e.  B  Y  =  [ q ] S
)
5 eropr.2 . . . . . . . 8  |-  K  =  ( B /. S
)
64, 5eleq2s 2179 . . . . . . 7  |-  ( Y  e.  K  ->  E. q  e.  B  Y  =  [ q ] S
)
73, 6anim12i 331 . . . . . 6  |-  ( ( X  e.  J  /\  Y  e.  K )  ->  ( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
87adantl 271 . . . . 5  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
9 reeanv 2532 . . . . 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 132 . . . 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 270 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  T  e.  Z )
13 ecexg 6248 . . . . . . 7  |-  ( T  e.  Z  ->  [ ( p  .+  q ) ] T  e.  _V )
14 elisset 2627 . . . . . . 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 298 . . . . 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 2399 . . . 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 145 . . 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 1831 . . . . . . . 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 130 . . . . . . 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 2381 . . . . . 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 2636 . . . . . 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 182 . . . . 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 2381 . . . 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 2636 . . . 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 182 . . 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 120 . 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 2532 . . . . . 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 6279 . . . . . . . . . . 11  |-  ( p  =  r  ->  [ p ] R  =  [
r ] R )
3029eqeq2d 2096 . . . . . . . . . 10  |-  ( p  =  r  ->  ( X  =  [ p ] R  <->  X  =  [
r ] R ) )
3130anbi1d 453 . . . . . . . . 9  |-  ( p  =  r  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
q ] S ) ) )
32 oveq1 5620 . . . . . . . . . . 11  |-  ( p  =  r  ->  (
p  .+  q )  =  ( r  .+  q ) )
3332eceq1d 6280 . . . . . . . . . 10  |-  ( p  =  r  ->  [ ( p  .+  q ) ] T  =  [
( r  .+  q
) ] T )
3433eqeq2d 2096 . . . . . . . . 9  |-  ( p  =  r  ->  (
z  =  [ ( p  .+  q ) ] T  <->  z  =  [ ( r  .+  q ) ] T
) )
3531, 34anbi12d 457 . . . . . . . 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 6279 . . . . . . . . . . 11  |-  ( q  =  t  ->  [ q ] S  =  [
t ] S )
3736eqeq2d 2096 . . . . . . . . . 10  |-  ( q  =  t  ->  ( Y  =  [ q ] S  <->  Y  =  [
t ] S ) )
3837anbi2d 452 . . . . . . . . 9  |-  ( q  =  t  ->  (
( X  =  [
r ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
t ] S ) ) )
39 oveq2 5621 . . . . . . . . . . 11  |-  ( q  =  t  ->  (
r  .+  q )  =  ( r  .+  t ) )
4039eceq1d 6280 . . . . . . . . . 10  |-  ( q  =  t  ->  [ ( r  .+  q ) ] T  =  [
( r  .+  t
) ] T )
4140eqeq2d 2096 . . . . . . . . 9  |-  ( q  =  t  ->  (
z  =  [ ( r  .+  q ) ] T  <->  z  =  [ ( r  .+  t ) ] T
) )
4238, 41anbi12d 457 . . . . . . . 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 2595 . . . . . . 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 6279 . . . . . . . . . . 11  |-  ( p  =  s  ->  [ p ] R  =  [
s ] R )
4544eqeq2d 2096 . . . . . . . . . 10  |-  ( p  =  s  ->  ( X  =  [ p ] R  <->  X  =  [
s ] R ) )
4645anbi1d 453 . . . . . . . . 9  |-  ( p  =  s  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
q ] S ) ) )
47 oveq1 5620 . . . . . . . . . . 11  |-  ( p  =  s  ->  (
p  .+  q )  =  ( s  .+  q ) )
4847eceq1d 6280 . . . . . . . . . 10  |-  ( p  =  s  ->  [ ( p  .+  q ) ] T  =  [
( s  .+  q
) ] T )
4948eqeq2d 2096 . . . . . . . . 9  |-  ( p  =  s  ->  (
w  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( s  .+  q ) ] T
) )
5046, 49anbi12d 457 . . . . . . . 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 6279 . . . . . . . . . . 11  |-  ( q  =  u  ->  [ q ] S  =  [
u ] S )
5251eqeq2d 2096 . . . . . . . . . 10  |-  ( q  =  u  ->  ( Y  =  [ q ] S  <->  Y  =  [
u ] S ) )
5352anbi2d 452 . . . . . . . . 9  |-  ( q  =  u  ->  (
( X  =  [
s ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
u ] S ) ) )
54 oveq2 5621 . . . . . . . . . . 11  |-  ( q  =  u  ->  (
s  .+  q )  =  ( s  .+  u ) )
5554eceq1d 6280 . . . . . . . . . 10  |-  ( q  =  u  ->  [ ( s  .+  q ) ] T  =  [
( s  .+  u
) ] T )
5655eqeq2d 2096 . . . . . . . . 9  |-  ( q  =  u  ->  (
w  =  [ ( s  .+  q ) ] T  <->  w  =  [ ( s  .+  u ) ] T
) )
5753, 56anbi12d 457 . . . . . . . 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 2595 . . . . . . 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 448 . . . . . 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 185 . . . . 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 2532 . . . . . . 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 270 . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  A  C_  U )
67 simprll 504 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  A )
6866, 67sseldd 3015 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  U )
6964, 68erth 6288 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  B  C_  V )
74 simprrl 506 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  B )
7573, 74sseldd 3015 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  V )
7671, 75erth 6288 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( t S u  <->  [ t ] S  =  [ u ] S
) )
7769, 76anbi12d 457 . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  .+  : ( A  X.  B ) --> C )
8483, 67, 74fovrnd 5746 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  C )
8581, 84sseldd 3015 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  W )
8679, 85erth 6288 . . . . . . . . . . . . . 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 200 . . . . . . . . . . . . 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 2094 . . . . . . . . . . . . . 14  |-  ( w  =  [ ( s 
.+  u ) ] T  ->  ( [
( r  .+  t
) ] T  =  w  <->  [ ( r  .+  t ) ] T  =  [ ( s  .+  u ) ] T
) )
8988biimprcd 158 . . . . . . . . . . . . 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 251 . . . . . . . . . . 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 2091 . . . . . . . . . . . . . . 15  |-  ( X  =  [ r ] R  ->  ( X  =  [ s ] R  <->  [ r ] R  =  [ s ] R
) )
93 eqeq1 2091 . . . . . . . . . . . . . . 15  |-  ( Y  =  [ t ] S  ->  ( Y  =  [ u ] S  <->  [ t ] S  =  [ u ] S
) )
9492, 93bi2anan9 571 . . . . . . . . . . . . . 14  |-  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  ->  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  <->  ( [ r ] R  =  [
s ] R  /\  [ t ] S  =  [ u ] S
) ) )
9594anbi1d 453 . . . . . . . . . . . . 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 270 . . . . . . . . . . . 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 2091 . . . . . . . . . . . . 13  |-  ( z  =  [ ( r 
.+  t ) ] T  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
9897adantl 271 . . . . . . . . . . . 12  |-  ( ( ( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
9996, 98imbi12d 232 . . . . . . . . . . 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 155 . . . . . . . . . 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 251 . . . . . . . . 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 392 . . . . . . . 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 2492 . . . . . . 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 151 . . . . . 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 2492 . . . . 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 151 . . . 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 270 . . 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 1800 . 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 2091 . . . . 5  |-  ( z  =  w  ->  (
z  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( p  .+  q ) ] T
) )
110109anbi2d 452 . . . 4  |-  ( z  =  w  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) ) )
1111102rexbidv 2399 . . 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 2007 . 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 408 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 102    <-> wb 103   A.wal 1285    = wceq 1287   E.wex 1424    e. wcel 1436   E!weu 1945   E.wrex 2356   _Vcvv 2615    C_ wss 2988   class class class wbr 3820    X. cxp 4409   -->wf 4977  (class class class)co 5613    Er wer 6241   [cec 6242   /.cqs 6243
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4094  df-xp 4417  df-rel 4418  df-cnv 4419  df-co 4420  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424  df-iota 4946  df-fun 4983  df-fn 4984  df-f 4985  df-fv 4989  df-ov 5616  df-er 6244  df-ec 6246  df-qs 6250
This theorem is referenced by:  erovlem  6336  eroprf  6337
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