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Theorem genpassl 7144
Description: Associativity of lower cuts. Lemma for genpassg 7146. (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
genpassl  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A F B ) F C ) )  =  ( 1st `  ( 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 genpassl
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 prop 7095 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 elprnql 7101 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
31, 2sylan 278 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
4 prop 7095 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
5 elprnql 7101 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
64, 5sylan 278 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
7 r19.41v 2524 . . . . . . . . . . . . . . . . 17  |-  ( E. h  e.  ( 1st `  C ) ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( E. h  e.  ( 1st `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) )
8 prop 7095 . . . . . . . . . . . . . . . . . . . . 21  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
9 elprnql 7101 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  h  e.  ( 1st `  C ) )  ->  h  e.  Q. )
108, 9sylan 278 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( C  e.  P.  /\  h  e.  ( 1st `  C ) )  ->  h  e.  Q. )
11 oveq2 5674 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  ( g G h )  ->  (
f G t )  =  ( f G ( g G h ) ) )
1211adantr 271 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f G t )  =  ( f G ( g G h ) ) )
13 genpassg.6 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
( f G g ) G h )  =  ( f G ( g G h ) ) )
1413adantl 272 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( (
f G g ) G h )  =  ( f G ( g G h ) ) )
1512, 14eqtr4d 2124 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f G t )  =  ( ( f G g ) G h ) )
1615eqeq2d 2100 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( x  =  ( f G t )  <->  x  =  ( ( f G g ) G h ) ) )
1716expcom 115 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  (
t  =  ( g G h )  -> 
( x  =  ( f G t )  <-> 
x  =  ( ( f G g ) G h ) ) ) )
1817pm5.32d 439 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
19183expa 1144 . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
2010, 19sylan2 281 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  ( C  e.  P.  /\  h  e.  ( 1st `  C ) ) )  ->  ( ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
2120anassrs 393 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( f  e. 
Q.  /\  g  e.  Q. )  /\  C  e. 
P. )  /\  h  e.  ( 1st `  C
) )  ->  (
( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
2221rexbidva 2378 . . . . . . . . . . . . . . . . 17  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  C  e.  P. )  ->  ( E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
237, 22syl5rbbr 194 . . . . . . . . . . . . . . . 16  |-  ( ( ( f  e.  Q.  /\  g  e.  Q. )  /\  C  e.  P. )  ->  ( E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 1st `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2423an32s 536 . . . . . . . . . . . . . . 15  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  g  e.  Q. )  ->  ( E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 1st `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
256, 24sylan2 281 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  ( B  e.  P.  /\  g  e.  ( 1st `  B ) ) )  ->  ( E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. h  e.  ( 1st `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2625anassrs 393 . . . . . . . . . . . . 13  |-  ( ( ( ( f  e. 
Q.  /\  C  e.  P. )  /\  B  e. 
P. )  /\  g  e.  ( 1st `  B
) )  ->  ( E. h  e.  ( 1st `  C ) ( t  =  ( g G h )  /\  x  =  ( (
f G g ) G h ) )  <-> 
( E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
2726rexbidva 2378 . . . . . . . . . . . 12  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. g  e.  ( 1st `  B
) ( E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
28 r19.41v 2524 . . . . . . . . . . . 12  |-  ( E. g  e.  ( 1st `  B ) ( E. h  e.  ( 1st `  C ) t  =  ( g G h )  /\  x  =  ( f G t ) )  <->  ( E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) )
2927, 28syl6bb 195 . . . . . . . . . . 11  |-  ( ( ( f  e.  Q.  /\  C  e.  P. )  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
3029an31s 538 . . . . . . . . . 10  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  ( E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
3130exbidv 1754 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C ) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. t
( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  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 5813 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
34 elisset 2634 . . . . . . . . . . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . . . . . . . . . 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 1831 . . . . . . . . . . . . . . . . . . . . 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, 37syl6bbr 197 . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
3910, 38sylan2 281 . . . . . . . . . . . . . . . . . . 19  |-  ( ( g  e.  Q.  /\  ( C  e.  P.  /\  h  e.  ( 1st `  C ) ) )  ->  ( x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4039anassrs 393 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( g  e.  Q.  /\  C  e.  P. )  /\  h  e.  ( 1st `  C ) )  ->  ( x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4140rexbidva 2378 . . . . . . . . . . . . . . . . 17  |-  ( ( g  e.  Q.  /\  C  e.  P. )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. h  e.  ( 1st `  C ) E. t ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
42 rexcom4 2643 . . . . . . . . . . . . . . . . 17  |-  ( E. h  e.  ( 1st `  C ) E. t
( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) )  <->  E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) )
4341, 42syl6bb 195 . . . . . . . . . . . . . . . 16  |-  ( ( g  e.  Q.  /\  C  e.  P. )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4443ancoms 265 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  P.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
456, 44sylan2 281 . . . . . . . . . . . . . 14  |-  ( ( C  e.  P.  /\  ( B  e.  P.  /\  g  e.  ( 1st `  B ) ) )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4645anassrs 393 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  P.  /\  B  e.  P. )  /\  g  e.  ( 1st `  B ) )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4746rexbidva 2378 . . . . . . . . . . . 12  |-  ( ( C  e.  P.  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 1st `  B ) E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
4847ancoms 265 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 1st `  B ) E. t E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
49 rexcom4 2643 . . . . . . . . . . 11  |-  ( E. g  e.  ( 1st `  B ) E. t E. h  e.  ( 1st `  C ) ( t  =  ( g G h )  /\  x  =  ( (
f G g ) G h ) )  <->  E. t E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) )
5048, 49syl6bb 195 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
5150adantr 271 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) ( t  =  ( g G h )  /\  x  =  ( ( f G g ) G h ) ) ) )
52 df-rex 2366 . . . . . . . . . . 11  |-  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. t ( t  e.  ( 1st `  ( 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, 32genpelvl 7132 . . . . . . . . . . . . 13  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( t  e.  ( 1st `  ( B F C ) )  <->  E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C ) t  =  ( g G h ) ) )
5554anbi1d 454 . . . . . . . . . . . 12  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( ( t  e.  ( 1st `  ( B F C ) )  /\  x  =  ( f G t ) )  <->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5655exbidv 1754 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. t ( t  e.  ( 1st `  ( B F C ) )  /\  x  =  ( f G t ) )  <->  E. t
( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5752, 56syl5bb 191 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. t
( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5857adantr 271 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. t
( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) t  =  ( g G h )  /\  x  =  ( f G t ) ) ) )
5931, 51, 583bitr4rd 220 . . . . . . . 8  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  f  e.  Q. )  ->  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
603, 59sylan2 281 . . . . . . 7  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  ( A  e.  P.  /\  f  e.  ( 1st `  A ) ) )  ->  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
6160anassrs 393 . . . . . 6  |-  ( ( ( ( B  e. 
P.  /\  C  e.  P. )  /\  A  e. 
P. )  /\  f  e.  ( 1st `  A
) )  ->  ( E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
6261rexbidva 2378 . . . . 5  |-  ( ( ( B  e.  P.  /\  C  e.  P. )  /\  A  e.  P. )  ->  ( E. f  e.  ( 1st `  A
) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
6362ancoms 265 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( E. f  e.  ( 1st `  A ) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
64633impb 1140 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. f  e.  ( 1st `  A ) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t )  <->  E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) )
65 genpassg.5 . . . . . 6  |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  ( f F g )  e.  P. )
6665caovcl 5813 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B F C )  e.  P. )
6753, 32genpelvl 7132 . . . . 5  |-  ( ( A  e.  P.  /\  ( B F C )  e.  P. )  -> 
( x  e.  ( 1st `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 1st `  A ) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t ) ) )
6866, 67sylan2 281 . . . 4  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  C  e.  P. )
)  ->  ( x  e.  ( 1st `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 1st `  A ) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t ) ) )
69683impb 1140 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  ( A F ( B F C ) ) )  <->  E. f  e.  ( 1st `  A
) E. t  e.  ( 1st `  ( B F C ) ) x  =  ( f G t ) ) )
70 df-rex 2366 . . . . 5  |-  ( E. t  e.  ( 1st `  ( A F B ) ) E. h  e.  ( 1st `  C
) x  =  ( t G h )  <->  E. t ( t  e.  ( 1st `  ( A F B ) )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
7153, 32genpelvl 7132 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( t  e.  ( 1st `  ( A F B ) )  <->  E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B ) t  =  ( f G g ) ) )
72713adant3 964 . . . . . . 7  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
t  e.  ( 1st `  ( A F B ) )  <->  E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g ) ) )
7372anbi1d 454 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
( t  e.  ( 1st `  ( A F B ) )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
7473exbidv 1754 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. t ( t  e.  ( 1st `  ( A F B ) )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  E. t ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
7570, 74syl5bb 191 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. t  e.  ( 1st `  ( A F B ) ) E. h  e.  ( 1st `  C ) x  =  ( t G h )  <->  E. t ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
7665caovcl 5813 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
7753, 32genpelvl 7132 . . . . . 6  |-  ( ( ( A F B )  e.  P.  /\  C  e.  P. )  ->  ( x  e.  ( 1st `  ( ( A F B ) F C ) )  <->  E. t  e.  ( 1st `  ( A F B ) ) E. h  e.  ( 1st `  C ) x  =  ( t G h ) ) )
7876, 77sylan 278 . . . . 5  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  C  e.  P. )  ->  ( x  e.  ( 1st `  (
( A F B ) F C ) )  <->  E. t  e.  ( 1st `  ( A F B ) ) E. h  e.  ( 1st `  C ) x  =  ( t G h ) ) )
79783impa 1139 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  ( ( A F B ) F C ) )  <->  E. t  e.  ( 1st `  ( A F B ) ) E. h  e.  ( 1st `  C ) x  =  ( t G h ) ) )
8032caovcl 5813 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f G g )  e.  Q. )
81 elisset 2634 . . . . . . . . . . . . . . . . . . 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 300 . . . . . . . . . . . . . . . . 17  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <-> 
( E. t  t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) ) )
84 oveq1 5673 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  ( f G g )  ->  (
t G h )  =  ( ( f G g ) G h ) )
8584eqeq2d 2100 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  ( f G g )  ->  (
x  =  ( t G h )  <->  x  =  ( ( f G g ) G h ) ) )
8685rexbidv 2382 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  ( f G g )  ->  ( E. h  e.  ( 1st `  C ) x  =  ( t G h )  <->  E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
8786pm5.32i 443 . . . . . . . . . . . . . . . . . . 19  |-  ( ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( t G h ) )  <->  ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) )
8887exbii 1542 . . . . . . . . . . . . . . . . . 18  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) )
89 19.41v 1831 . . . . . . . . . . . . . . . . . 18  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) )  <->  ( E. t 
t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) )
9088, 89bitri 183 . . . . . . . . . . . . . . . . 17  |-  ( E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  ( E. t 
t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( ( f G g ) G h ) ) )
9183, 90syl6bbr 197 . . . . . . . . . . . . . . . 16  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
926, 91sylan2 281 . . . . . . . . . . . . . . 15  |-  ( ( f  e.  Q.  /\  ( B  e.  P.  /\  g  e.  ( 1st `  B ) ) )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
9392anassrs 393 . . . . . . . . . . . . . 14  |-  ( ( ( f  e.  Q.  /\  B  e.  P. )  /\  g  e.  ( 1st `  B ) )  ->  ( E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
9493rexbidva 2378 . . . . . . . . . . . . 13  |-  ( ( f  e.  Q.  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. g  e.  ( 1st `  B ) E. t ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
95 rexcom4 2643 . . . . . . . . . . . . 13  |-  ( E. g  e.  ( 1st `  B ) E. t
( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
9694, 95syl6bb 195 . . . . . . . . . . . 12  |-  ( ( f  e.  Q.  /\  B  e.  P. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
9796ancoms 265 . . . . . . . . . . 11  |-  ( ( B  e.  P.  /\  f  e.  Q. )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
983, 97sylan2 281 . . . . . . . . . 10  |-  ( ( B  e.  P.  /\  ( A  e.  P.  /\  f  e.  ( 1st `  A ) ) )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
9998anassrs 393 . . . . . . . . 9  |-  ( ( ( B  e.  P.  /\  A  e.  P. )  /\  f  e.  ( 1st `  A ) )  ->  ( E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
10099rexbidva 2378 . . . . . . . 8  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. f  e.  ( 1st `  A ) E. t E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
101 rexcom4 2643 . . . . . . . 8  |-  ( E. f  e.  ( 1st `  A ) E. t E. g  e.  ( 1st `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( t G h ) )  <->  E. t E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( t G h ) ) )
102100, 101syl6bb 195 . . . . . . 7  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
103 r19.41v 2524 . . . . . . . . . 10  |-  ( E. g  e.  ( 1st `  B ) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C ) x  =  ( t G h ) )  <->  ( E. g  e.  ( 1st `  B ) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
104103rexbii 2386 . . . . . . . . 9  |-  ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  E. f  e.  ( 1st `  A ) ( E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
105 r19.41v 2524 . . . . . . . . 9  |-  ( E. f  e.  ( 1st `  A ) ( E. g  e.  ( 1st `  B ) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
106104, 105bitri 183 . . . . . . . 8  |-  ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
107106exbii 1542 . . . . . . 7  |-  ( E. t E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) ( t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) )  <->  E. t ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) )
108102, 107syl6bb 195 . . . . . 6  |-  ( ( B  e.  P.  /\  A  e.  P. )  ->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
109108ancoms 265 . . . . 5  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
1101093adant3 964 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( E. f  e.  ( 1st `  A ) E. g  e.  ( 1st `  B ) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h )  <->  E. t ( E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) t  =  ( f G g )  /\  E. h  e.  ( 1st `  C
) x  =  ( t G h ) ) ) )
11175, 79, 1103bitr4d 219 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  ( ( A F B ) F C ) )  <->  E. f  e.  ( 1st `  A
) E. g  e.  ( 1st `  B
) E. h  e.  ( 1st `  C
) x  =  ( ( f G g ) G h ) ) )
11264, 69, 1113bitr4rd 220 . 2  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  (
x  e.  ( 1st `  ( ( A F B ) F C ) )  <->  x  e.  ( 1st `  ( A F ( B F C ) ) ) ) )
113112eqrdv 2087 1  |-  ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  ( ( A F B ) F C ) )  =  ( 1st `  ( A F ( B F C ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    = wceq 1290   E.wex 1427    e. wcel 1439   E.wrex 2361   {crab 2364   <.cop 3453    X. cxp 4450   dom cdm 4452   ` cfv 5028  (class class class)co 5666    |-> cmpt2 5668   1stc1st 5923   2ndc2nd 5924   Q.cnq 6900   P.cnp 6911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-coll 3960  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-rab 2369  df-v 2622  df-sbc 2842  df-csb 2935  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926  df-qs 6312  df-ni 6924  df-nqqs 6968  df-inp 7086
This theorem is referenced by:  genpassg  7146
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