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Theorem genpassu 7515
Description: Associativity of upper cuts. Lemma for genpassg 7516. (Contributed by Jim Kingdon, 11-Dec-2019.)
Hypotheses
Ref Expression
genpelvl.1  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
genpelvl.2  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
genpassg.4  |-  dom  F  =  ( P.  X.  P. )
genpassg.5  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  e.  P. )
genpassg.6  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f G g ) G h )  =  ( f G ( g G h ) ) )
Assertion
Ref Expression
genpassu  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) )
Distinct variable groups:    x, y, z, f, g, h, w, v, A    x, B, y, z, f, g, h, w, v    x, G, y, z, f, g, h, w, v    f, F, g    C, f, g, h, v, w, x, y, z    h, F, v, w, x, y, z

Proof of Theorem genpassu
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 prop 7465 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 elprnqu 7472 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
31, 2sylan 283 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  ( 2nd `  A ) )  -> 
f  e.  Q. )
4 prop 7465 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
5 elprnqu 7472 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
64, 5sylan 283 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  g  e.  ( 2nd `  B ) )  -> 
g  e.  Q. )
7 prop 7465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
8 elprnqu 7472 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  h  e.  ( 2nd `  C ) )  ->  h  e.  Q. )
97, 8sylan 283 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( C  e.  P.  /\  h  e.  ( 2nd `  C ) )  ->  h  e.  Q. )
10 oveq2 5877 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  ( g G h )  ->  (
f G t )  =  ( f G ( g G h ) ) )
1110adantr 276 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f G t )  =  ( f G ( g G h ) ) )
12 genpassg.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f G g ) G h )  =  ( f G ( g G h ) ) )
1312adantl 277 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( (
f G g ) G h )  =  ( f G ( g G h ) ) )
1411, 13eqtr4d 2213 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f G t )  =  ( ( f G g ) G h ) )
1514eqeq2d 2189 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( x  =  ( f G t )  <->  x  =  ( ( f G g ) G h ) ) )
1615expcom 116 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
t  =  ( g G h )  -> 
( x  =  ( f G t )  <-> 
x  =  ( ( f G g ) G h ) ) ) )
1716pm5.32d 450 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
18173expa 1203 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  h  e.  Q. )  ->  ( ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
199, 18sylan2 286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  ( C  e.  P.  /\  h  e.  ( 2nd `  C ) ) )  ->  ( ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
2019anassrs 400 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( f  e. 
Q.  /\  g  e.  Q. )  /\  C  e. 
P. )  /\  h  e.  ( 2nd `  C
) )  ->  (
( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
2120rexbidva 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  C  e.  P. )  ->  ( E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
22 r19.41v 2633 . . . . . . . . . . . . . . . . 17  |-  ( E. h  e.  ( 2nd `  C ) ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( E. h  e.  ( 2nd `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) )
2321, 22bitr3di 195 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  C  e.  P. )  ->  ( E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 2nd `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2423an32s 568 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  g  e.  Q. )  ->  ( E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 2nd `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
256, 24sylan2 286 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 2nd `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2625anassrs 400 . . . . . . . . . . . . 13  |-  ( ( ( ( f  e. 
Q.  /\  C  e.  P. )  /\  B  e. 
P. )  /\  g  e.  ( 2nd `  B
) )  ->  ( E. h  e.  ( 2nd `  C ) ( t  =  ( g G h )  /\  x  =  ( (
f G g ) G h ) )  <-> 
( E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2726rexbidva 2474 . . . . . . . . . . . 12  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. g  e.  ( 2nd `  B
) ( E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
28 r19.41v 2633 . . . . . . . . . . . 12  |-  ( E. g  e.  ( 2nd `  B ) ( E. h  e.  ( 2nd `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) )
2927, 28bitrdi 196 . . . . . . . . . . 11  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
3029an31s 570 . . . . . . . . . 10  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
3130exbidv 1825 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C ) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. t
( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
32 genpelvl.2 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )
3332caovcl 6023 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
34 elisset 2751 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( g G h )  e.  Q.  ->  E. t 
t  =  ( g G h ) )
3533, 34syl 14 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  E. t  t  =  ( g G h ) )
3635biantrurd 305 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( x  =  ( ( f G g ) G h )  <-> 
( E. t  t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
37 19.41v 1902 . . . . . . . . . . . . . . . . . . . . 21  |-  ( E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. t  t  =  (
g G h )  /\  x  =  ( ( f G g ) G h ) ) )
3836, 37bitr4di 198 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
399, 38sylan2 286 . . . . . . . . . . . . . . . . . . 19  |-  ( ( g  e.  Q.  /\  ( C  e.  P.  /\  h  e.  ( 2nd `  C ) ) )  ->  ( x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4039anassrs 400 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( g  e.  Q.  /\  C  e.  P. )  /\  h  e.  ( 2nd `  C ) )  ->  ( x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4140rexbidva 2474 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  Q.  /\  C  e.  P. )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. h  e.  ( 2nd `  C ) E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
42 rexcom4 2760 . . . . . . . . . . . . . . . . 17  |-  ( E. h  e.  ( 2nd `  C ) E. t
( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) )
4341, 42bitrdi 196 . . . . . . . . . . . . . . . 16  |-  ( ( g  e.  Q.  /\  C  e.  P. )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4443ancoms 268 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  P.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
456, 44sylan2 286 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4645anassrs 400 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  P.  /\  B  e.  P. )  /\  g  e.  ( 2nd `  B ) )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4746rexbidva 2474 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 2nd `  B ) E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4847ancoms 268 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 2nd `  B ) E. t E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
49 rexcom4 2760 . . . . . . . . . . 11  |-  ( E. g  e.  ( 2nd `  B ) E. t E. h  e.  ( 2nd `  C ) ( t  =  ( g G h )  /\  x  =  ( (
f G g ) G h ) )  <->  E. t E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) )
5048, 49bitrdi 196 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
5150adantr 276 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
52 df-rex 2461 . . . . . . . . . . 11  |-  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. t ( t  e.  ( 2nd `  ( B F C ) )  /\  x  =  ( f G t ) ) )
53 genpelvl.1 . . . . . . . . . . . . . 14  |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y G z ) ) } >. )
5453, 32genpelvu 7503 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( t  e.  ( 2nd `  ( B F C ) )  <->  E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C ) t  =  ( g G h ) ) )
5554anbi1d 465 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( t  e.  ( 2nd `  ( B F C ) )  /\  x  =  ( f G t ) )  <->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5655exbidv 1825 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. t ( t  e.  ( 2nd `  ( B F C ) )  /\  x  =  ( f G t ) )  <->  E. t
( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5752, 56bitrid 192 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. t
( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5857adantr 276 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. t
( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5931, 51, 583bitr4rd 221 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
603, 59sylan2 286 . . . . . . 7  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( A  e.  P.  /\  f  e.  ( 2nd `  A ) ) )  ->  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
6160anassrs 400 . . . . . 6  |-  ( ( ( ( B  e. 
P.  /\  C  e.  P. )  /\  A  e. 
P. )  /\  f  e.  ( 2nd `  A
) )  ->  ( E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
6261rexbidva 2474 . . . . 5  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  A  e.  P. )  ->  ( E. f  e.  ( 2nd `  A
) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
6362ancoms 268 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( E. f  e.  ( 2nd `  A ) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
64633impb 1199 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. f  e.  ( 2nd `  A ) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) )
65 genpassg.5 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  e.  P. )
6665caovcl 6023 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B F C )  e.  P. )
6753, 32genpelvu 7503 . . . . 5  |-  ( ( A  e.  P.  /\  ( B F C )  e.  P. )  -> 
( x  e.  ( 2nd `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 2nd `  A ) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t ) ) )
6866, 67sylan2 286 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( x  e.  ( 2nd `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 2nd `  A ) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t ) ) )
69683impb 1199 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 2nd `  A
) E. t  e.  ( 2nd `  ( B F C ) ) x  =  ( f G t ) ) )
70 df-rex 2461 . . . . 5  |-  ( E. t  e.  ( 2nd `  ( A F B ) ) E. h  e.  ( 2nd `  C
) x  =  ( t G h )  <->  E. t ( t  e.  ( 2nd `  ( A F B ) )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
7153, 32genpelvu 7503 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( t  e.  ( 2nd `  ( A F B ) )  <->  E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B ) t  =  ( f G g ) ) )
72713adant3 1017 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
t  e.  ( 2nd `  ( A F B ) )  <->  E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g ) ) )
7372anbi1d 465 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( t  e.  ( 2nd `  ( A F B ) )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
7473exbidv 1825 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. t ( t  e.  ( 2nd `  ( A F B ) )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  E. t ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
7570, 74bitrid 192 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. t  e.  ( 2nd `  ( A F B ) ) E. h  e.  ( 2nd `  C ) x  =  ( t G h )  <->  E. t ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
7665caovcl 6023 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
7753, 32genpelvu 7503 . . . . . 6  |-  ( ( ( A F B )  e.  P.  /\  C  e.  P. )  ->  ( x  e.  ( 2nd `  ( ( A F B ) F C ) )  <->  E. t  e.  ( 2nd `  ( A F B ) ) E. h  e.  ( 2nd `  C ) x  =  ( t G h ) ) )
7876, 77sylan 283 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  P. )  ->  ( x  e.  ( 2nd `  (
( A F B ) F C ) )  <->  E. t  e.  ( 2nd `  ( A F B ) ) E. h  e.  ( 2nd `  C ) x  =  ( t G h ) ) )
79783impa 1194 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  ( ( A F B ) F C ) )  <->  E. t  e.  ( 2nd `  ( A F B ) ) E. h  e.  ( 2nd `  C ) x  =  ( t G h ) ) )
8032caovcl 6023 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f G g )  e.  Q. )
81 elisset 2751 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f G g )  e.  Q.  ->  E. t 
t  =  ( f G g ) )
8280, 81syl 14 . . . . . . . . . . . . . . . . . 18  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  E. t  t  =  ( f G g ) )
8382biantrurd 305 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <-> 
( E. t  t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) ) )
84 oveq1 5876 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  ( f G g )  ->  (
t G h )  =  ( ( f G g ) G h ) )
8584eqeq2d 2189 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  ( f G g )  ->  (
x  =  ( t G h )  <->  x  =  ( ( f G g ) G h ) ) )
8685rexbidv 2478 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  ( f G g )  ->  ( E. h  e.  ( 2nd `  C ) x  =  ( t G h )  <->  E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
8786pm5.32i 454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( t G h ) )  <->  ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) )
8887exbii 1605 . . . . . . . . . . . . . . . . . 18  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) )
89 19.41v 1902 . . . . . . . . . . . . . . . . . 18  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) )  <->  ( E. t 
t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) )
9088, 89bitri 184 . . . . . . . . . . . . . . . . 17  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  ( E. t 
t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( ( f G g ) G h ) ) )
9183, 90bitr4di 198 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
926, 91sylan2 286 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  ( B  e.  P.  /\  g  e.  ( 2nd `  B ) ) )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
9392anassrs 400 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  Q.  /\  B  e.  P. )  /\  g  e.  ( 2nd `  B ) )  ->  ( E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
9493rexbidva 2474 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 2nd `  B ) E. t ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
95 rexcom4 2760 . . . . . . . . . . . . 13  |-  ( E. g  e.  ( 2nd `  B ) E. t
( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
9694, 95bitrdi 196 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  B  e.  P. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
9796ancoms 268 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  f  e.  Q. )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
983, 97sylan2 286 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  ( A  e.  P.  /\  f  e.  ( 2nd `  A ) ) )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
9998anassrs 400 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  A  e.  P. )  /\  f  e.  ( 2nd `  A ) )  ->  ( E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
10099rexbidva 2474 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. f  e.  ( 2nd `  A ) E. t E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
101 rexcom4 2760 . . . . . . . 8  |-  ( E. f  e.  ( 2nd `  A ) E. t E. g  e.  ( 2nd `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( t G h ) )  <->  E. t E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( t G h ) ) )
102100, 101bitrdi 196 . . . . . . 7  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
103 r19.41v 2633 . . . . . . . . . 10  |-  ( E. g  e.  ( 2nd `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C ) x  =  ( t G h ) )  <->  ( E. g  e.  ( 2nd `  B ) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
104103rexbii 2484 . . . . . . . . 9  |-  ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  E. f  e.  ( 2nd `  A ) ( E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
105 r19.41v 2633 . . . . . . . . 9  |-  ( E. f  e.  ( 2nd `  A ) ( E. g  e.  ( 2nd `  B ) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
106104, 105bitri 184 . . . . . . . 8  |-  ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
107106exbii 1605 . . . . . . 7  |-  ( E. t E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) )  <->  E. t ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) )
108102, 107bitrdi 196 . . . . . 6  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
109108ancoms 268 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
1101093adant3 1017 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. f  e.  ( 2nd `  A ) E. g  e.  ( 2nd `  B ) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) t  =  ( f G g )  /\  E. h  e.  ( 2nd `  C
) x  =  ( t G h ) ) ) )
11175, 79, 1103bitr4d 220 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  ( ( A F B ) F C ) )  <->  E. f  e.  ( 2nd `  A
) E. g  e.  ( 2nd `  B
) E. h  e.  ( 2nd `  C
) x  =  ( ( f G g ) G h ) ) )
11264, 69, 1113bitr4rd 221 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 2nd `  ( ( A F B ) F C ) )  <->  x  e.  ( 2nd `  ( A F ( B F C ) ) ) ) )
113112eqrdv 2175 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  ( ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    /\ w3a 978    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3594    X. cxp 4621   dom cdm 4623   ` cfv 5212  (class class class)co 5869    e. cmpo 5871   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   P.cnp 7281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-qs 6535  df-ni 7294  df-nqqs 7338  df-inp 7456
This theorem is referenced by:  genpassg  7516
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