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Theorem genpassl 7486
Description: Associativity of lower cuts. Lemma for genpassg 7488. (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 7437 . . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 elprnql 7443 . . . . . . . . 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 7437 . . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
5 elprnql 7443 . . . . . . . . . . . . . . . 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 prop 7437 . . . . . . . . . . . . . . . . . . . . 21 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
8 elprnql 7443 . . . . . . . . . . . . . . . . . . . . 21 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ h e. ( 1st ` C ) ) -> h e. Q. )
97, 8sylan 281 . . . . . . . . . . . . . . . . . . . 20 |- ( ( C e. P. /\ h e. ( 1st ` C ) ) -> h e. Q. )
10 oveq2 5861 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( t = ( g G h ) -> ( f G t ) = ( f G ( g G h ) ) )
1110adantr 274 . . . . . . . . . . . . . . . . . . . . . . . . 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 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 ) ) )
1411, 13eqtr4d 2206 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f G t ) = ( ( f G g ) G h ) )
1514eqeq2d 2182 . . . . . . . . . . . . . . . . . . . . . . 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 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 ) ) ) )
1716pm5.32d 447 . . . . . . . . . . . . . . . . . . . . 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 1198 . . . . . . . . . . . . . . . . . . . 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 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 ) ) ) )
2019anassrs 398 . . . . . . . . . . . . . . . . . 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 ) ) ) )
2120rexbidva 2467 . . . . . . . . . . . . . . . . 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 ) ) ) )
22 r19.41v 2626 . . . . . . . . . . . . . . . . 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 ) ) )
2321, 22bitr3di 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 563 . . . . . . . . . . . . . . 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 398 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 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 2626 . . . . . . . . . . . 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, 28bitrdi 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 565 . . . . . . . . . 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 1818 . . . . . . . . 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 6007 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
34 elisset 2744 . . . . . . . . . . . . . . . . . . . . . . 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 1895 . . . . . . . . . . . . . . . . . . . . 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 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 ) ) ) )
399, 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 398 . . . . . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . . . . . 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 2753 . . . . . . . . . . . . . . . . 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, 42bitrdi 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 398 . . . . . . . . . . . . 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 2467 . . . . . . . . . . . 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 2753 . . . . . . . . . . 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, 49bitrdi 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 2454 . . . . . . . . . . 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 7474 . . . . . . . . . . . . 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 462 . . . . . . . . . . . 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 1818 . . . . . . . . . . 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 398 . . . . . 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 2467 . . . . 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 1194 . . 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 6007 . . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
6753, 32genpelvl 7474 . . . . 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 1194 . . 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 2454 . . . . 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 7474 . . . . . . . 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 1012 . . . . . . 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 462 . . . . . 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 1818 . . . . 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 6007 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
7753, 32genpelvl 7474 . . . . . 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 1189 . . . 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 6007 . . . . . . . . . . . . . . . . . . 19 |- ( ( f e. Q. /\ g e. Q. ) -> ( f G g ) e. Q. )
81 elisset 2744 . . . . . . . . . . . . . . . . . . 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 5860 . . . . . . . . . . . . . . . . . . . . . 22 |- ( t = ( f G g ) -> ( t G h ) = ( ( f G g ) G h ) )
8584eqeq2d 2182 . . . . . . . . . . . . . . . . . . . . 21 |- ( t = ( f G g ) -> ( x = ( t G h ) <-> x = ( ( f G g ) G h ) ) )
8685rexbidv 2471 . . . . . . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . . . . 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 1598 . . . . . . . . . . . . . . . . . 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 1895 . . . . . . . . . . . . . . . . . 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, 90bitr4di 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 398 . . . . . . . . . . . . . 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 2467 . . . . . . . . . . . . 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 2753 . . . . . . . . . . . . 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, 95bitrdi 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 398 . . . . . . . . 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 2467 . . . . . . . 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 2753 . . . . . . . 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, 101bitrdi 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 2626 . . . . . . . . . 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 2477 . . . . . . . . 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 2626 . . . . . . . . 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 1598 . . . . . . 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, 107bitrdi 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 1012 . . . 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 2168 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 973   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   {crab 2452   <.cop 3586   X. cxp 4609   dom cdm 4611   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   P.cnp 7253
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-qs 6519  df-ni 7266  df-nqqs 7310  df-inp 7428
This theorem is referenced by:  genpassg  7488
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