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Theorem erovlem 6364
Description: Lemma for eroprf 6365. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 30-Dec-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 ) ) )
eropr.12  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) }
Assertion
Ref Expression
erovlem  |-  ( ph  -> 
.+^  =  ( x  e.  J ,  y  e.  K  |->  ( iota 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, x, y, z, A    B, p, q, r, s, t, u, x, y, z    J, p, q, x, y, z    R, p, q, r, s, t, u, x, y, z    K, p, q, x, y, z    S, p, q, r, s, t, u, x, y, z    .+ , p, q, r, s, t, u, x, y, z    ph, p, q, r, s, t, u, x, y, z    T, p, q, r, s, t, u, x, y, z
Allowed substitution hints:    C( x, y, z, u, t, s, r, q, p)    .+^ ( x, y, z, u, t, s, r, q, p)    U( x, y, z, u, t, s, r, q, p)    J( u, t, s, r)    K( u, t, s, r)    V( x, y, z, u, t, s, r, q, p)    W( x, y, z, u, t, s, r, q, p)    Z( x, y, z, u, t, s, r, q, p)

Proof of Theorem erovlem
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 simpl 107 . . . . . . . 8  |-  ( ( ( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  ->  ( x  =  [ p ] R  /\  y  =  [
q ] S ) )
21reximi 2470 . . . . . . 7  |-  ( E. q  e.  B  ( ( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  ->  E. q  e.  B  ( x  =  [ p ] R  /\  y  =  [
q ] S ) )
32reximi 2470 . . . . . 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 ) )
4 eropr.1 . . . . . . . . . 10  |-  J  =  ( A /. R
)
54eleq2i 2154 . . . . . . . . 9  |-  ( x  e.  J  <->  x  e.  ( A /. R ) )
6 vex 2622 . . . . . . . . . 10  |-  x  e. 
_V
76elqs 6323 . . . . . . . . 9  |-  ( x  e.  ( A /. R )  <->  E. p  e.  A  x  =  [ p ] R
)
85, 7bitri 182 . . . . . . . 8  |-  ( x  e.  J  <->  E. p  e.  A  x  =  [ p ] R
)
9 eropr.2 . . . . . . . . . 10  |-  K  =  ( B /. S
)
109eleq2i 2154 . . . . . . . . 9  |-  ( y  e.  K  <->  y  e.  ( B /. S ) )
11 vex 2622 . . . . . . . . . 10  |-  y  e. 
_V
1211elqs 6323 . . . . . . . . 9  |-  ( y  e.  ( B /. S )  <->  E. q  e.  B  y  =  [ q ] S
)
1310, 12bitri 182 . . . . . . . 8  |-  ( y  e.  K  <->  E. q  e.  B  y  =  [ q ] S
)
148, 13anbi12i 448 . . . . . . 7  |-  ( ( x  e.  J  /\  y  e.  K )  <->  ( E. p  e.  A  x  =  [ p ] R  /\  E. q  e.  B  y  =  [ q ] S
) )
15 reeanv 2536 . . . . . . 7  |-  ( 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 ) )
1614, 15bitr4i 185 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  K )  <->  E. p  e.  A  E. q  e.  B  (
x  =  [ p ] R  /\  y  =  [ q ] S
) )
173, 16sylibr 132 . . . . 5  |-  ( E. p  e.  A  E. q  e.  B  (
( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  ->  ( x  e.  J  /\  y  e.  K ) )
1817pm4.71ri 384 . . . 4  |-  ( E. p  e.  A  E. q  e.  B  (
( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  <->  ( ( x  e.  J  /\  y  e.  K )  /\  E. p  e.  A  E. q  e.  B  (
( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
) ) )
19 eropr.3 . . . . . . . 8  |-  ( ph  ->  T  e.  Z )
20 eropr.4 . . . . . . . 8  |-  ( ph  ->  R  Er  U )
21 eropr.5 . . . . . . . 8  |-  ( ph  ->  S  Er  V )
22 eropr.6 . . . . . . . 8  |-  ( ph  ->  T  Er  W )
23 eropr.7 . . . . . . . 8  |-  ( ph  ->  A  C_  U )
24 eropr.8 . . . . . . . 8  |-  ( ph  ->  B  C_  V )
25 eropr.9 . . . . . . . 8  |-  ( ph  ->  C  C_  W )
26 eropr.10 . . . . . . . 8  |-  ( ph  ->  .+  : ( A  X.  B ) --> C )
27 eropr.11 . . . . . . . 8  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r R s  /\  t S u )  ->  (
r  .+  t ) T ( s  .+  u ) ) )
284, 9, 19, 20, 21, 22, 23, 24, 25, 26, 27eroveu 6363 . . . . . . 7  |-  ( (
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 ) )
29 iota1 4981 . . . . . . 7  |-  ( E! z 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
)  /\  z  =  [ ( p  .+  q ) ] T
)  <->  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )  =  z ) )
3028, 29syl 14 . . . . . 6  |-  ( (
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 )  <-> 
( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  =  z ) )
31 eqcom 2090 . . . . . 6  |-  ( ( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  =  z  <-> 
z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )
3230, 31syl6bb 194 . . . . 5  |-  ( (
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 )  <-> 
z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) )
3332pm5.32da 440 . . . 4  |-  ( 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
) )  <->  ( (
x  e.  J  /\  y  e.  K )  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) ) )
3418, 33syl5bb 190 . . 3  |-  ( ph  ->  ( E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( x  e.  J  /\  y  e.  K )  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) ) )
3534oprabbidv 5685 . 2  |-  ( ph  ->  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) }  =  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  J  /\  y  e.  K
)  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) ) } )
36 eropr.12 . 2  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) }
37 df-mpt2 5639 . . 3  |-  ( x  e.  J ,  y  e.  K  |->  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )  =  { <. <. x ,  y
>. ,  w >.  |  ( ( x  e.  J  /\  y  e.  K )  /\  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) }
38 nfv 1466 . . . 4  |-  F/ w
( ( x  e.  J  /\  y  e.  K )  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )
39 nfv 1466 . . . . 5  |-  F/ z ( x  e.  J  /\  y  e.  K
)
40 nfiota1 4969 . . . . . 6  |-  F/_ z
( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )
4140nfeq2 2240 . . . . 5  |-  F/ z  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )
4239, 41nfan 1502 . . . 4  |-  F/ z ( ( x  e.  J  /\  y  e.  K )  /\  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )
43 eqeq1 2094 . . . . 5  |-  ( z  =  w  ->  (
z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) )  <->  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) )
4443anbi2d 452 . . . 4  |-  ( z  =  w  ->  (
( ( x  e.  J  /\  y  e.  K )  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )  <->  ( (
x  e.  J  /\  y  e.  K )  /\  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) ) )
4538, 42, 44cbvoprab3 5706 . . 3  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  J  /\  y  e.  K
)  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) ) }  =  { <. <. x ,  y >. ,  w >.  |  ( ( x  e.  J  /\  y  e.  K )  /\  w  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) }
4637, 45eqtr4i 2111 . 2  |-  ( x  e.  J ,  y  e.  K  |->  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) )  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  J  /\  y  e.  K )  /\  z  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) ) ) }
4735, 36, 463eqtr4g 2145 1  |-  ( ph  -> 
.+^  =  ( x  e.  J ,  y  e.  K  |->  ( iota 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    = wceq 1289    e. wcel 1438   E!weu 1948   E.wrex 2360    C_ wss 2997   class class class wbr 3837    X. cxp 4426   iotacio 4965   -->wf 4998  (class class class)co 5634   {coprab 5635    |-> cmpt2 5636    Er wer 6269   [cec 6270   /.cqs 6271
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-er 6272  df-ec 6274  df-qs 6278
This theorem is referenced by:  eroprf  6365
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