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Theorem eroveu 7000
Description: Lemma for erov 7002 and eroprf 7003. (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 6959 . . . . . . . 8  |-  ( X  e.  ( A /. R )  ->  E. p  e.  A  X  =  [ p ] R
)
2 eropr.1 . . . . . . . 8  |-  J  =  ( A /. R
)
31, 2eleq2s 2529 . . . . . . 7  |-  ( X  e.  J  ->  E. p  e.  A  X  =  [ p ] R
)
4 elqsi 6959 . . . . . . . 8  |-  ( Y  e.  ( B /. S )  ->  E. q  e.  B  Y  =  [ q ] S
)
5 eropr.2 . . . . . . . 8  |-  K  =  ( B /. S
)
64, 5eleq2s 2529 . . . . . . 7  |-  ( Y  e.  K  ->  E. q  e.  B  Y  =  [ q ] S
)
73, 6anim12i 551 . . . . . 6  |-  ( ( X  e.  J  /\  Y  e.  K )  ->  ( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
87adantl 454 . . . . 5  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  -> 
( E. p  e.  A  X  =  [
p ] R  /\  E. q  e.  B  Y  =  [ q ] S
) )
9 reeanv 2876 . . . . 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 205 . . . 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 453 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  T  e.  Z )
13 ecexg 6910 . . . . . . 7  |-  ( T  e.  Z  ->  [ ( p  .+  q ) ] T  e.  _V )
14 elisset 2967 . . . . . . 7  |-  ( [ ( p  .+  q
) ] T  e. 
_V  ->  E. z  z  =  [ ( p  .+  q ) ] T
)
1512, 13, 143syl 19 . . . . . 6  |-  ( (
ph  /\  ( X  e.  J  /\  Y  e.  K ) )  ->  E. z  z  =  [ ( p  .+  q ) ] T
)
1615biantrud 495 . . . . 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 2749 . . . 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 203 . . 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 1929 . . . . . . . 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 195 . . . . . . 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 2731 . . . . . 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 2976 . . . . . 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 242 . . . . 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 2731 . . . 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 2976 . . . 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 242 . . 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 190 . 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 2876 . . . . . 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 6942 . . . . . . . . . . 11  |-  ( p  =  r  ->  [ p ] R  =  [
r ] R )
3029eqeq2d 2448 . . . . . . . . . 10  |-  ( p  =  r  ->  ( X  =  [ p ] R  <->  X  =  [
r ] R ) )
3130anbi1d 687 . . . . . . . . 9  |-  ( p  =  r  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
q ] S ) ) )
32 oveq1 6089 . . . . . . . . . . 11  |-  ( p  =  r  ->  (
p  .+  q )  =  ( r  .+  q ) )
33 eceq1 6942 . . . . . . . . . . 11  |-  ( ( p  .+  q )  =  ( r  .+  q )  ->  [ ( p  .+  q ) ] T  =  [
( r  .+  q
) ] T )
3432, 33syl 16 . . . . . . . . . 10  |-  ( p  =  r  ->  [ ( p  .+  q ) ] T  =  [
( r  .+  q
) ] T )
3534eqeq2d 2448 . . . . . . . . 9  |-  ( p  =  r  ->  (
z  =  [ ( p  .+  q ) ] T  <->  z  =  [ ( r  .+  q ) ] T
) )
3631, 35anbi12d 693 . . . . . . . 8  |-  ( p  =  r  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ r ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( r  .+  q
) ] T ) ) )
37 eceq1 6942 . . . . . . . . . . 11  |-  ( q  =  t  ->  [ q ] S  =  [
t ] S )
3837eqeq2d 2448 . . . . . . . . . 10  |-  ( q  =  t  ->  ( Y  =  [ q ] S  <->  Y  =  [
t ] S ) )
3938anbi2d 686 . . . . . . . . 9  |-  ( q  =  t  ->  (
( X  =  [
r ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ r ] R  /\  Y  =  [
t ] S ) ) )
40 oveq2 6090 . . . . . . . . . . 11  |-  ( q  =  t  ->  (
r  .+  q )  =  ( r  .+  t ) )
41 eceq1 6942 . . . . . . . . . . 11  |-  ( ( r  .+  q )  =  ( r  .+  t )  ->  [ ( r  .+  q ) ] T  =  [
( r  .+  t
) ] T )
4240, 41syl 16 . . . . . . . . . 10  |-  ( q  =  t  ->  [ ( r  .+  q ) ] T  =  [
( r  .+  t
) ] T )
4342eqeq2d 2448 . . . . . . . . 9  |-  ( q  =  t  ->  (
z  =  [ ( r  .+  q ) ] T  <->  z  =  [ ( r  .+  t ) ] T
) )
4439, 43anbi12d 693 . . . . . . . 8  |-  ( q  =  t  ->  (
( ( X  =  [ r ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( r  .+  q
) ] T )  <-> 
( ( X  =  [ r ] R  /\  Y  =  [
t ] S )  /\  z  =  [
( r  .+  t
) ] T ) ) )
4536, 44cbvrex2v 2942 . . . . . . 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 ) )
46 eceq1 6942 . . . . . . . . . . 11  |-  ( p  =  s  ->  [ p ] R  =  [
s ] R )
4746eqeq2d 2448 . . . . . . . . . 10  |-  ( p  =  s  ->  ( X  =  [ p ] R  <->  X  =  [
s ] R ) )
4847anbi1d 687 . . . . . . . . 9  |-  ( p  =  s  ->  (
( X  =  [
p ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
q ] S ) ) )
49 oveq1 6089 . . . . . . . . . . 11  |-  ( p  =  s  ->  (
p  .+  q )  =  ( s  .+  q ) )
50 eceq1 6942 . . . . . . . . . . 11  |-  ( ( p  .+  q )  =  ( s  .+  q )  ->  [ ( p  .+  q ) ] T  =  [
( s  .+  q
) ] T )
5149, 50syl 16 . . . . . . . . . 10  |-  ( p  =  s  ->  [ ( p  .+  q ) ] T  =  [
( s  .+  q
) ] T )
5251eqeq2d 2448 . . . . . . . . 9  |-  ( p  =  s  ->  (
w  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( s  .+  q ) ] T
) )
5348, 52anbi12d 693 . . . . . . . 8  |-  ( p  =  s  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ s ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( s  .+  q
) ] T ) ) )
54 eceq1 6942 . . . . . . . . . . 11  |-  ( q  =  u  ->  [ q ] S  =  [
u ] S )
5554eqeq2d 2448 . . . . . . . . . 10  |-  ( q  =  u  ->  ( Y  =  [ q ] S  <->  Y  =  [
u ] S ) )
5655anbi2d 686 . . . . . . . . 9  |-  ( q  =  u  ->  (
( X  =  [
s ] R  /\  Y  =  [ q ] S )  <->  ( X  =  [ s ] R  /\  Y  =  [
u ] S ) ) )
57 oveq2 6090 . . . . . . . . . . 11  |-  ( q  =  u  ->  (
s  .+  q )  =  ( s  .+  u ) )
58 eceq1 6942 . . . . . . . . . . 11  |-  ( ( s  .+  q )  =  ( s  .+  u )  ->  [ ( s  .+  q ) ] T  =  [
( s  .+  u
) ] T )
5957, 58syl 16 . . . . . . . . . 10  |-  ( q  =  u  ->  [ ( s  .+  q ) ] T  =  [
( s  .+  u
) ] T )
6059eqeq2d 2448 . . . . . . . . 9  |-  ( q  =  u  ->  (
w  =  [ ( s  .+  q ) ] T  <->  w  =  [ ( s  .+  u ) ] T
) )
6156, 60anbi12d 693 . . . . . . . 8  |-  ( q  =  u  ->  (
( ( X  =  [ s ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( s  .+  q
) ] T )  <-> 
( ( X  =  [ s ] R  /\  Y  =  [
u ] S )  /\  w  =  [
( s  .+  u
) ] T ) ) )
6253, 61cbvrex2v 2942 . . . . . . 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 ) )
6345, 62anbi12i 680 . . . . . 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
) ) )
6428, 63bitr4i 245 . . . . 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
) ) )
65 reeanv 2876 . . . . . . 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
) ) )
66 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 ) ) )
67 eropr.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  R  Er  U )
6867adantr 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  R  Er  U )
69 eropr.7 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A  C_  U )
7069adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  A  C_  U )
71 simprll 740 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  A )
7270, 71sseldd 3350 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
r  e.  U )
7368, 72erth 6950 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r R s  <->  [ r ] R  =  [ s ] R
) )
74 eropr.5 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  S  Er  V )
7574adantr 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  S  Er  V )
76 eropr.8 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  B  C_  V )
7776adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  B  C_  V )
78 simprrl 742 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  B )
7977, 78sseldd 3350 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
t  e.  V )
8075, 79erth 6950 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( t S u  <->  [ t ] S  =  [ u ] S
) )
8173, 80anbi12d 693 . . . . . . . . . . . . . 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
) ) )
82 eropr.6 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  T  Er  W )
8382adantr 453 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  T  Er  W )
84 eropr.9 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  C  C_  W )
8584adantr 453 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  C  C_  W )
86 eropr.10 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  .+  : ( A  X.  B ) --> C )
8786adantr 453 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  ->  .+  : ( A  X.  B ) --> C )
8887, 71, 78fovrnd 6219 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  C )
8985, 88sseldd 3350 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( r  .+  t
)  e.  W )
9083, 89erth 6950 . . . . . . . . . . . . . 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
) )
9166, 81, 903imtr3d 260 . . . . . . . . . . . . 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 ) )
92 eqeq2 2446 . . . . . . . . . . . . . 14  |-  ( w  =  [ ( s 
.+  u ) ] T  ->  ( [
( r  .+  t
) ] T  =  w  <->  [ ( r  .+  t ) ] T  =  [ ( s  .+  u ) ] T
) )
9392biimprcd 218 . . . . . . . . . . . . 13  |-  ( [ ( r  .+  t
) ] T  =  [ ( s  .+  u ) ] T  ->  ( w  =  [
( s  .+  u
) ] T  ->  [ ( r  .+  t ) ] T  =  w ) )
9491, 93syl6 32 . . . . . . . . . . . 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 ) ) )
9594imp3a 422 . . . . . . . . . . 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
) )
96 eqeq1 2443 . . . . . . . . . . . . . . 15  |-  ( X  =  [ r ] R  ->  ( X  =  [ s ] R  <->  [ r ] R  =  [ s ] R
) )
97 eqeq1 2443 . . . . . . . . . . . . . . 15  |-  ( Y  =  [ t ] S  ->  ( Y  =  [ u ] S  <->  [ t ] S  =  [ u ] S
) )
9896, 97bi2anan9 845 . . . . . . . . . . . . . 14  |-  ( ( X  =  [ r ] R  /\  Y  =  [ t ] S
)  ->  ( ( X  =  [ s ] R  /\  Y  =  [ u ] S
)  <->  ( [ r ] R  =  [
s ] R  /\  [ t ] S  =  [ u ] S
) ) )
9998anbi1d 687 . . . . . . . . . . . . 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
) ) )
10099adantr 453 . . . . . . . . . . . 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
) ) )
101 eqeq1 2443 . . . . . . . . . . . . 13  |-  ( z  =  [ ( r 
.+  t ) ] T  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
102101adantl 454 . . . . . . . . . . . 12  |-  ( ( ( X  =  [
r ] R  /\  Y  =  [ t ] S )  /\  z  =  [ ( r  .+  t ) ] T
)  ->  ( z  =  w  <->  [ ( r  .+  t ) ] T  =  w ) )
103100, 102imbi12d 313 . . . . . . . . . . 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
) ) )
10495, 103syl5ibrcom 215 . . . . . . . . . 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 ) ) )
105104imp3a 422 . . . . . . . . 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 ) )
106105anassrs 631 . . . . . . . 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 ) )
107106rexlimdvva 2838 . . . . . . 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 ) )
10865, 107syl5bir 211 . . . . . 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 ) )
109108rexlimdvva 2838 . . . . 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 ) )
11064, 109syl5bir 211 . . . 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 ) )
111110adantr 453 . . 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 ) )
112111alrimivv 1643 . 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 ) )
113 eqeq1 2443 . . . . 5  |-  ( z  =  w  ->  (
z  =  [ ( p  .+  q ) ] T  <->  w  =  [ ( p  .+  q ) ] T
) )
114113anbi2d 686 . . . 4  |-  ( z  =  w  ->  (
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( X  =  [ p ] R  /\  Y  =  [
q ] S )  /\  w  =  [
( p  .+  q
) ] T ) ) )
1151142rexbidv 2749 . . 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 ) ) )
116115eu4 2321 . 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 ) ) )
11727, 112, 116sylanbrc 647 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    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551    = wceq 1653    e. wcel 1726   E!weu 2282   E.wrex 2707   _Vcvv 2957    C_ wss 3321   class class class wbr 4213    X. cxp 4877   -->wf 5451  (class class class)co 6082    Er wer 6903   [cec 6904   /.cqs 6905
This theorem is referenced by:  erovlem  7001  erov  7002  eroprf  7003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-er 6906  df-ec 6908  df-qs 6912
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