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Theorem erovlem 6754
Description: Lemma for erov 6755 and eroprf 6756. (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 443 . . . . . . . 8  |-  ( ( ( x  =  [
p ] R  /\  y  =  [ q ] S )  /\  z  =  [ ( p  .+  q ) ] T
)  ->  ( x  =  [ p ] R  /\  y  =  [
q ] S ) )
21reximi 2650 . . . . . . 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 2650 . . . . . 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 2347 . . . . . . . . 9  |-  ( x  e.  J  <->  x  e.  ( A /. R ) )
6 vex 2791 . . . . . . . . . 10  |-  x  e. 
_V
76elqs 6712 . . . . . . . . 9  |-  ( x  e.  ( A /. R )  <->  E. p  e.  A  x  =  [ p ] R
)
85, 7bitri 240 . . . . . . . 8  |-  ( x  e.  J  <->  E. p  e.  A  x  =  [ p ] R
)
9 eropr.2 . . . . . . . . . 10  |-  K  =  ( B /. S
)
109eleq2i 2347 . . . . . . . . 9  |-  ( y  e.  K  <->  y  e.  ( B /. S ) )
11 vex 2791 . . . . . . . . . 10  |-  y  e. 
_V
1211elqs 6712 . . . . . . . . 9  |-  ( y  e.  ( B /. S )  <->  E. q  e.  B  y  =  [ q ] S
)
1310, 12bitri 240 . . . . . . . 8  |-  ( y  e.  K  <->  E. q  e.  B  y  =  [ q ] S
)
148, 13anbi12i 678 . . . . . . 7  |-  ( ( x  e.  J  /\  y  e.  K )  <->  ( E. p  e.  A  x  =  [ p ] R  /\  E. q  e.  B  y  =  [ q ] S
) )
15 reeanv 2707 . . . . . . 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 243 . . . . . 6  |-  ( ( x  e.  J  /\  y  e.  K )  <->  E. p  e.  A  E. q  e.  B  (
x  =  [ p ] R  /\  y  =  [ q ] S
) )
173, 16sylibr 203 . . . . 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 614 . . . 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 6753 . . . . . . 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 5233 . . . . . . 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 15 . . . . . 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 2285 . . . . . 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 252 . . . . 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 622 . . . 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 248 . . 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 5902 . 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 5863 . . 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 1605 . . . 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 1605 . . . . 5  |-  F/ z ( x  e.  J  /\  y  e.  K
)
40 nfiota1 5221 . . . . . 6  |-  F/_ z
( iota z E. p  e.  A  E. q  e.  B  ( (
x  =  [ p ] R  /\  y  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )
4140nfeq2 2430 . . . . 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 1771 . . . 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 2289 . . . . 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 684 . . . 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 5922 . . 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 2306 . 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 2340 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   E.wrex 2544    C_ wss 3152   class class class wbr 4023    X. cxp 4687   iotacio 5217   -->wf 5251  (class class class)co 5858   {coprab 5859    e. cmpt2 5860    Er wer 6657   [cec 6658   /.cqs 6659
This theorem is referenced by:  erov  6755  eroprf  6756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-ec 6662  df-qs 6666
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