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Theorem erov 6771
Description: The value of an operation defined on equivalence classes. (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 ) }
eropr.13  |-  ( ph  ->  R  e.  X )
eropr.14  |-  ( ph  ->  S  e.  Y )
Assertion
Ref Expression
erov  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( [ P ] R  .+^  [ Q ] S )  =  [
( 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    P, p, q, r, s, t, u, x, y, z    R, p, q, r, s, t, u, x, y, z    K, p, q, x, y, z    Q, p, q, r, s, t, u, 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    X, p, q, r, s, t, u, z    Y, p, q, r, s, t, u, 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)    X( x, y)    Y( x, y)    Z( x, y, z, u, t, s, r, q, p)

Proof of Theorem erov
StepHypRef Expression
1 eropr.1 . . . . 5  |-  J  =  ( A /. R
)
2 eropr.2 . . . . 5  |-  K  =  ( B /. S
)
3 eropr.3 . . . . 5  |-  ( ph  ->  T  e.  Z )
4 eropr.4 . . . . 5  |-  ( ph  ->  R  Er  U )
5 eropr.5 . . . . 5  |-  ( ph  ->  S  Er  V )
6 eropr.6 . . . . 5  |-  ( ph  ->  T  Er  W )
7 eropr.7 . . . . 5  |-  ( ph  ->  A  C_  U )
8 eropr.8 . . . . 5  |-  ( ph  ->  B  C_  V )
9 eropr.9 . . . . 5  |-  ( ph  ->  C  C_  W )
10 eropr.10 . . . . 5  |-  ( ph  ->  .+  : ( A  X.  B ) --> C )
11 eropr.11 . . . . 5  |-  ( (
ph  /\  ( (
r  e.  A  /\  s  e.  A )  /\  ( t  e.  B  /\  u  e.  B
) ) )  -> 
( ( r R s  /\  t S u )  ->  (
r  .+  t ) T ( s  .+  u ) ) )
12 eropr.12 . . . . 5  |-  .+^  =  { <. <. x ,  y
>. ,  z >.  |  E. p  e.  A  E. q  e.  B  ( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T ) }
131, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12erovlem 6770 . . . 4  |-  ( 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 ) ) ) )
14133ad2ant1 976 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  .+^  =  ( 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
) ) ) )
15 simprl 732 . . . . . . . 8  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  x  =  [ P ] R
)
1615eqeq1d 2304 . . . . . . 7  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  (
x  =  [ p ] R  <->  [ P ] R  =  [ p ] R
) )
17 simprr 733 . . . . . . . 8  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  y  =  [ Q ] S
)
1817eqeq1d 2304 . . . . . . 7  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  (
y  =  [ q ] S  <->  [ Q ] S  =  [
q ] S ) )
1916, 18anbi12d 691 . . . . . 6  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  (
( x  =  [
p ] R  /\  y  =  [ q ] S )  <->  ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
) ) )
2019anbi1d 685 . . . . 5  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  (
( ( x  =  [ p ] R  /\  y  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) )
21202rexbidv 2599 . . . 4  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  ( 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  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) )
2221iotabidv 5256 . . 3  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  (
x  =  [ P ] R  /\  y  =  [ Q ] S
) )  ->  ( iota z 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  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) )
23 eropr.13 . . . . 5  |-  ( ph  ->  R  e.  X )
24 ecelqsg 6730 . . . . . 6  |-  ( ( R  e.  X  /\  P  e.  A )  ->  [ P ] R  e.  ( A /. R
) )
2524, 1syl6eleqr 2387 . . . . 5  |-  ( ( R  e.  X  /\  P  e.  A )  ->  [ P ] R  e.  J )
2623, 25sylan 457 . . . 4  |-  ( (
ph  /\  P  e.  A )  ->  [ P ] R  e.  J
)
27263adant3 975 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  [ P ] R  e.  J
)
28 eropr.14 . . . . 5  |-  ( ph  ->  S  e.  Y )
29 ecelqsg 6730 . . . . . 6  |-  ( ( S  e.  Y  /\  Q  e.  B )  ->  [ Q ] S  e.  ( B /. S
) )
3029, 2syl6eleqr 2387 . . . . 5  |-  ( ( S  e.  Y  /\  Q  e.  B )  ->  [ Q ] S  e.  K )
3128, 30sylan 457 . . . 4  |-  ( (
ph  /\  Q  e.  B )  ->  [ Q ] S  e.  K
)
32313adant2 974 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  [ Q ] S  e.  K
)
33 iotaex 5252 . . . 4  |-  ( iota z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  e.  _V
3433a1i 10 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( iota z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  e.  _V )
3514, 22, 27, 32, 34ovmpt2d 5991 . 2  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( [ P ] R  .+^  [ Q ] S )  =  ( iota z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) ) )
36 eqid 2296 . . . . . . 7  |-  [ P ] R  =  [ P ] R
37 eqid 2296 . . . . . . 7  |-  [ Q ] S  =  [ Q ] S
3836, 37pm3.2i 441 . . . . . 6  |-  ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ Q ] S
)
39 eqid 2296 . . . . . 6  |-  [ ( P  .+  Q ) ] T  =  [
( P  .+  Q
) ] T
4038, 39pm3.2i 441 . . . . 5  |-  ( ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ Q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( P  .+  Q ) ] T )
41 eceq1 6712 . . . . . . . . 9  |-  ( p  =  P  ->  [ p ] R  =  [ P ] R )
4241eqeq2d 2307 . . . . . . . 8  |-  ( p  =  P  ->  ( [ P ] R  =  [ p ] R  <->  [ P ] R  =  [ P ] R
) )
4342anbi1d 685 . . . . . . 7  |-  ( p  =  P  ->  (
( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [
q ] S )  <-> 
( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [
q ] S ) ) )
44 oveq1 5881 . . . . . . . . 9  |-  ( p  =  P  ->  (
p  .+  q )  =  ( P  .+  q ) )
45 eceq1 6712 . . . . . . . . 9  |-  ( ( p  .+  q )  =  ( P  .+  q )  ->  [ ( p  .+  q ) ] T  =  [
( P  .+  q
) ] T )
4644, 45syl 15 . . . . . . . 8  |-  ( p  =  P  ->  [ ( p  .+  q ) ] T  =  [
( P  .+  q
) ] T )
4746eqeq2d 2307 . . . . . . 7  |-  ( p  =  P  ->  ( [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T  <->  [ ( P  .+  Q
) ] T  =  [ ( P  .+  q ) ] T
) )
4843, 47anbi12d 691 . . . . . 6  |-  ( p  =  P  ->  (
( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T )  <->  ( ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( P  .+  q ) ] T ) ) )
49 eceq1 6712 . . . . . . . . 9  |-  ( q  =  Q  ->  [ q ] S  =  [ Q ] S )
5049eqeq2d 2307 . . . . . . . 8  |-  ( q  =  Q  ->  ( [ Q ] S  =  [ q ] S  <->  [ Q ] S  =  [ Q ] S
) )
5150anbi2d 684 . . . . . . 7  |-  ( q  =  Q  ->  (
( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [
q ] S )  <-> 
( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ Q ] S ) ) )
52 oveq2 5882 . . . . . . . . 9  |-  ( q  =  Q  ->  ( P  .+  q )  =  ( P  .+  Q
) )
53 eceq1 6712 . . . . . . . . 9  |-  ( ( P  .+  q )  =  ( P  .+  Q )  ->  [ ( P  .+  q ) ] T  =  [
( P  .+  Q
) ] T )
5452, 53syl 15 . . . . . . . 8  |-  ( q  =  Q  ->  [ ( P  .+  q ) ] T  =  [
( P  .+  Q
) ] T )
5554eqeq2d 2307 . . . . . . 7  |-  ( q  =  Q  ->  ( [ ( P  .+  Q ) ] T  =  [ ( P  .+  q ) ] T  <->  [ ( P  .+  Q
) ] T  =  [ ( P  .+  Q ) ] T
) )
5651, 55anbi12d 691 . . . . . 6  |-  ( q  =  Q  ->  (
( ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [
q ] S )  /\  [ ( P 
.+  Q ) ] T  =  [ ( P  .+  q ) ] T )  <->  ( ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ Q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( P  .+  Q ) ] T ) ) )
5748, 56rspc2ev 2905 . . . . 5  |-  ( ( P  e.  A  /\  Q  e.  B  /\  ( ( [ P ] R  =  [ P ] R  /\  [ Q ] S  =  [ Q ] S )  /\  [ ( P  .+  Q
) ] T  =  [ ( P  .+  Q ) ] T
) )  ->  E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T ) )
5840, 57mp3an3 1266 . . . 4  |-  ( ( P  e.  A  /\  Q  e.  B )  ->  E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T ) )
59583adant1 973 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T ) )
60 ecexg 6680 . . . . . 6  |-  ( T  e.  Z  ->  [ ( P  .+  Q ) ] T  e.  _V )
613, 60syl 15 . . . . 5  |-  ( ph  ->  [ ( P  .+  Q ) ] T  e.  _V )
62613ad2ant1 976 . . . 4  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  [ ( P  .+  Q ) ] T  e.  _V )
63 simp1 955 . . . . 5  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ph )
641, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11eroveu 6769 . . . . 5  |-  ( (
ph  /\  ( [ P ] R  e.  J  /\  [ Q ] S  e.  K ) )  ->  E! z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )
6563, 27, 32, 64syl12anc 1180 . . . 4  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  E! z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )
66 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  z  =  [ ( P  .+  Q ) ] T
)  ->  z  =  [ ( P  .+  Q ) ] T
)
6766eqeq1d 2304 . . . . . 6  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  z  =  [ ( P  .+  Q ) ] T
)  ->  ( z  =  [ ( p  .+  q ) ] T  <->  [ ( P  .+  Q
) ] T  =  [ ( p  .+  q ) ] T
) )
6867anbi2d 684 . . . . 5  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  z  =  [ ( P  .+  Q ) ] T
)  ->  ( (
( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <-> 
( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T ) ) )
69682rexbidv 2599 . . . 4  |-  ( ( ( ph  /\  P  e.  A  /\  Q  e.  B )  /\  z  =  [ ( P  .+  Q ) ] T
)  ->  ( E. p  e.  A  E. q  e.  B  (
( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [
q ] S )  /\  z  =  [
( p  .+  q
) ] T )  <->  E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  [ ( P  .+  Q ) ] T  =  [ ( p  .+  q ) ] T ) ) )
7062, 65, 69iota2d 5260 . . 3  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( E. p  e.  A  E. q  e.  B  (
( [ P ] R  =  [ p ] R  /\  [ Q ] S  =  [
q ] S )  /\  [ ( P 
.+  Q ) ] T  =  [ ( p  .+  q ) ] T )  <->  ( iota z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  =  [
( P  .+  Q
) ] T ) )
7159, 70mpbid 201 . 2  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( iota z E. p  e.  A  E. q  e.  B  ( ( [ P ] R  =  [
p ] R  /\  [ Q ] S  =  [ q ] S
)  /\  z  =  [ ( p  .+  q ) ] T
) )  =  [
( P  .+  Q
) ] T )
7235, 71eqtrd 2328 1  |-  ( (
ph  /\  P  e.  A  /\  Q  e.  B
)  ->  ( [ P ] R  .+^  [ Q ] S )  =  [
( P  .+  Q
) ] T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   E!weu 2156   E.wrex 2557   _Vcvv 2801    C_ wss 3165   class class class wbr 4039    X. cxp 4703   iotacio 5233   -->wf 5267  (class class class)co 5874   {coprab 5875    e. cmpt2 5876    Er wer 6673   [cec 6674   /.cqs 6675
This theorem is referenced by:  erov2  6773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-ec 6678  df-qs 6682
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