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Theorem genpassl 7325
Description: Associativity of lower cuts. Lemma for genpassg 7327. (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 7276 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
2 elprnql 7282 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
31, 2sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  f  e.  ( 1st `  A ) )  -> 
f  e.  Q. )
4 prop 7276 . . . . . . . . . . . . . . . 16  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
5 elprnql 7282 . . . . . . . . . . . . . . . 16  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
64, 5sylan 281 . . . . . . . . . . . . . . 15  |-  ( ( B  e.  P.  /\  g  e.  ( 1st `  B ) )  -> 
g  e.  Q. )
7 r19.41v 2585 . . . . . . . . . . . . . . . . 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 7276 . . . . . . . . . . . . . . . . . . . . 21  |-  ( C  e.  P.  ->  <. ( 1st `  C ) ,  ( 2nd `  C
) >.  e.  P. )
9 elprnql 7282 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
<. ( 1st `  C
) ,  ( 2nd `  C ) >.  e.  P.  /\  h  e.  ( 1st `  C ) )  ->  h  e.  Q. )
108, 9sylan 281 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( C  e.  P.  /\  h  e.  ( 1st `  C ) )  ->  h  e.  Q. )
11 oveq2 5775 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  ( g G h )  ->  (
f G t )  =  ( f G ( g G h ) ) )
1211adantr 274 . . . . . . . . . . . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . . . . . . . . . . . . . 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 2173 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( t  =  ( g G h )  /\  ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )
)  ->  ( f G t )  =  ( ( f G g ) G h ) )
1615eqeq2d 2149 . . . . . . . . . . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . . . . . . . . . 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 1181 . . . . . . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . 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 2432 . . . . . . . . . . . . . . . . 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 557 . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . 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 2432 . . . . . . . . . . . 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 2585 . . . . . . . . . . . 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 559 . . . . . . . . . 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 1797 . . . . . . . . 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 5918 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( g  e.  Q.  /\  h  e.  Q. )  ->  ( g G h )  e.  Q. )
34 elisset 2695 . . . . . . . . . . . . . . . . . . . . . . 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 303 . . . . . . . . . . . . . . . . . . . . 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 1874 . . . . . . . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . . . . . 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 2432 . . . . . . . . . . . . . . . . 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 2704 . . . . . . . . . . . . . . . . 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 266 . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . 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 2432 . . . . . . . . . . . 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 266 . . . . . . . . . . 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 2704 . . . . . . . . . . 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 274 . . . . . . . . 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 2420 . . . . . . . . . . 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 7313 . . . . . . . . . . . . 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 460 . . . . . . . . . . . 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 1797 . . . . . . . . . . 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 274 . . . . . . . . 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 284 . . . . . . 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 397 . . . . . 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 2432 . . . . 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 266 . . . 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 1177 . . 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 5918 . . . . 5  |-  ( ( B  e.  P.  /\  C  e.  P. )  ->  ( B F C )  e.  P. )
6753, 32genpelvl 7313 . . . . 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 284 . . . 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 1177 . . 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 2420 . . . . 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 7313 . . . . . . . 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 1001 . . . . . . 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 460 . . . . . 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 1797 . . . . 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 5918 . . . . . 6  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e.  P. )
7753, 32genpelvl 7313 . . . . . 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 281 . . . . 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 1176 . . . 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 5918 . . . . . . . . . . . . . . . . . . 19  |-  ( ( f  e.  Q.  /\  g  e.  Q. )  ->  ( f G g )  e.  Q. )
81 elisset 2695 . . . . . . . . . . . . . . . . . . 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 303 . . . . . . . . . . . . . . . . 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 5774 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  ( f G g )  ->  (
t G h )  =  ( ( f G g ) G h ) )
8584eqeq2d 2149 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  ( f G g )  ->  (
x  =  ( t G h )  <->  x  =  ( ( f G g ) G h ) ) )
8685rexbidv 2436 . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . 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 1584 . . . . . . . . . . . . . . . . . 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 1874 . . . . . . . . . . . . . . . . . 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 284 . . . . . . . . . . . . . . 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 397 . . . . . . . . . . . . . 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 2432 . . . . . . . . . . . . 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 2704 . . . . . . . . . . . . 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 266 . . . . . . . . . . 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 284 . . . . . . . . . 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 397 . . . . . . . . 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 2432 . . . . . . . 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 2704 . . . . . . . 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 2585 . . . . . . . . . 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 2440 . . . . . . . . 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 2585 . . . . . . . . 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 1584 . . . . . . 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 266 . . . . 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 1001 . . . 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 2135 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 962    = wceq 1331   E.wex 1468    e. wcel 1480   E.wrex 2415   {crab 2418   <.cop 3525    X. cxp 4532   dom cdm 4534   ` cfv 5118  (class class class)co 5767    e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   P.cnp 7092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267
This theorem is referenced by:  genpassg  7327
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