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Theorem genpassl 7465
Description: Associativity of lower cuts. Lemma for genpassg 7467. (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 7416 . . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2 elprnql 7422 . . . . . . . . 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 7416 . . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
5 elprnql 7422 . . . . . . . . . . . . . . . 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 7416 . . . . . . . . . . . . . . . . . . . . 21 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
8 elprnql 7422 . . . . . . . . . . . . . . . . . . . . 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 5850 . . . . . . . . . . . . . . . . . . . . . . . . . 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 2201 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f G t ) = ( ( f G g ) G h ) )
1514eqeq2d 2177 . . . . . . . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . . 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 558 . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . 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 2622 . . . . . . . . . . . 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 560 . . . . . . . . . 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 1813 . . . . . . . . 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 5996 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
34 elisset 2740 . . . . . . . . . . . . . . . . . . . . . . 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 1890 . . . . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . 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 2749 . . . . . . . . . . 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 2450 . . . . . . . . . . 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 7453 . . . . . . . . . . . . 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 461 . . . . . . . . . . . 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 1813 . . . . . . . . . . 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 2463 . . . . 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 1189 . . 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 5996 . . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
6753, 32genpelvl 7453 . . . . 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 1189 . . 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 2450 . . . . 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 7453 . . . . . . . 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 1007 . . . . . . 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 461 . . . . . 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 1813 . . . . 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 5996 . . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
7753, 32genpelvl 7453 . . . . . 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 1184 . . . 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 5996 . . . . . . . . . . . . . . . . . . 19 |- ( ( f e. Q. /\ g e. Q. ) -> ( f G g ) e. Q. )
81 elisset 2740 . . . . . . . . . . . . . . . . . . 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 5849 . . . . . . . . . . . . . . . . . . . . . 22 |- ( t = ( f G g ) -> ( t G h ) = ( ( f G g ) G h ) )
8584eqeq2d 2177 . . . . . . . . . . . . . . . . . . . . 21 |- ( t = ( f G g ) -> ( x = ( t G h ) <-> x = ( ( f G g ) G h ) ) )
8685rexbidv 2467 . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . 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 1593 . . . . . . . . . . . . . . . . . 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 1890 . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . 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 2749 . . . . . . . . . . . . 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 2463 . . . . . . . 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 2749 . . . . . . . 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 2622 . . . . . . . . . 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 2473 . . . . . . . . 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 2622 . . . . . . . . 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 1593 . . . . . . 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 1007 . . . 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 2163 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 968   = wceq 1343   E.wex 1480   e. wcel 2136   E.wrex 2445   {crab 2448   <.cop 3579   X. cxp 4602   dom cdm 4604   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   Q.cnq 7221   P.cnp 7232
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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565
This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-qs 6507  df-ni 7245  df-nqqs 7289  df-inp 7407
This theorem is referenced by:  genpassg  7467
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