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Theorem erovlem 6992
Description: Lemma for erov 6993 and eroprf 6994. (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 444 . . . . . . . 8  |-  ( ( ( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  ->  ( x  =  [ p ] R  /\  y  =  [
q ] S ) )
21reximi 2805 . . . . . . 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 2805 . . . . . 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 2499 . . . . . . . . 9  |-  ( x  e.  J  <->  x  e.  ( A /. R ) )
6 vex 2951 . . . . . . . . . 10  |-  x  e. 
_V
76elqs 6949 . . . . . . . . 9  |-  ( x  e.  ( A /. R )  <->  E. p  e.  A  x  =  [ p ] R
)
85, 7bitri 241 . . . . . . . 8  |-  ( x  e.  J  <->  E. p  e.  A  x  =  [ p ] R
)
9 eropr.2 . . . . . . . . . 10  |-  K  =  ( B /. S
)
109eleq2i 2499 . . . . . . . . 9  |-  ( y  e.  K  <->  y  e.  ( B /. S ) )
11 vex 2951 . . . . . . . . . 10  |-  y  e. 
_V
1211elqs 6949 . . . . . . . . 9  |-  ( y  e.  ( B /. S )  <->  E. q  e.  B  y  =  [ q ] S
)
1310, 12bitri 241 . . . . . . . 8  |-  ( y  e.  K  <->  E. q  e.  B  y  =  [ q ] S
)
148, 13anbi12i 679 . . . . . . 7  |-  ( ( x  e.  J  /\  y  e.  K )  <->  ( E. p  e.  A  x  =  [ p ] R  /\  E. q  e.  B  y  =  [ q ] S
) )
15 reeanv 2867 . . . . . . 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 244 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  K )  <->  E. p  e.  A  E. q  e.  B  (
x  =  [ p ] R  /\  y  =  [ q ] S
) )
173, 16sylibr 204 . . . . 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 615 . . . 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 6991 . . . . . . 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 5424 . . . . . . 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 16 . . . . . 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 2437 . . . . . 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 253 . . . . 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 623 . . . 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 249 . . 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 6120 . 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 6078 . . 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 1629 . . . 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 1629 . . . . 5  |-  F/ z ( x  e.  J  /\  y  e.  K
)
40 nfiota1 5412 . . . . . 6  |-  F/_ z
( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )
4140nfeq2 2582 . . . . 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 1846 . . . 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 2441 . . . . 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 685 . . . 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 6140 . . 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 2458 . 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 2492 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    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E!weu 2280   E.wrex 2698    C_ wss 3312   class class class wbr 4204    X. cxp 4868   iotacio 5408   -->wf 5442  (class class class)co 6073   {coprab 6074    e. cmpt2 6075    Er wer 6894   [cec 6895   /.cqs 6896
This theorem is referenced by:  erov  6993  eroprf  6994
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-er 6897  df-ec 6899  df-qs 6903
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