Theorem genpassu 7524

Description: Associativity of upper cuts. Lemma for genpassg 7525. (Contributed by Jim Kingdon, 11-Dec-2019.)

Hypotheses

Ref	Expression
genpelvl.1	|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
genpelvl.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
genpassg.4	|- dom F = ( P. X. P. )
genpassg.5	|- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )
genpassg.6	|- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )

Assertion

Ref	Expression
genpassu	|- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) )

Distinct variable groups:   x, y, z, f, g, h, w, v, A   x, B, y, z, f, g, h, w, v   x, G, y, z, f, g, h, w, v   f, F, g   C, f, g, h, v, w, x, y, z   h, F, v, w, x, y, z

Proof of Theorem genpassu

Dummy variable t is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7474	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		elprnqu 7481	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
3	1, 2	sylan 283	. . . . . . . 8 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
4		prop 7474	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
5		elprnqu 7481	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
6	4, 5	sylan 283	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
7		prop 7474	. . . . . . . . . . . . . . . . . . . . 21 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
8		elprnqu 7481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ h e. ( 2nd ` C ) ) -> h e. Q. )
9	7, 8	sylan 283	. . . . . . . . . . . . . . . . . . . 20 |- ( ( C e. P. /\ h e. ( 2nd ` C ) ) -> h e. Q. )
10		oveq2 5883	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( t = ( g G h ) -> ( f G t ) = ( f G ( g G h ) ) )
11	10	adantr 276	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f G t ) = ( f G ( g G h ) ) )
12		genpassg.6	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )
13	12	adantl 277	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )
14	11, 13	eqtr4d 2213	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f G t ) = ( ( f G g ) G h ) )
15	14	eqeq2d 2189	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( x = ( f G t ) <-> x = ( ( f G g ) G h ) ) )
16	15	expcom 116	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( t = ( g G h ) -> ( x = ( f G t ) <-> x = ( ( f G g ) G h ) ) ) )
17	16	pm5.32d 450	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( t = ( g G h ) /\ x = ( f G t ) ) <-> ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
18	17	3expa 1203	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( f e. Q. /\ g e. Q. ) /\ h e. Q. ) -> ( ( t = ( g G h ) /\ x = ( f G t ) ) <-> ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
19	9, 18	sylan2 286	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( f e. Q. /\ g e. Q. ) /\ ( C e. P. /\ h e. ( 2nd ` C ) ) ) -> ( ( t = ( g G h ) /\ x = ( f G t ) ) <-> ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
20	19	anassrs 400	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( f e. Q. /\ g e. Q. ) /\ C e. P. ) /\ h e. ( 2nd ` C ) ) -> ( ( t = ( g G h ) /\ x = ( f G t ) ) <-> ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
21	20	rexbidva 2474	. . . . . . . . . . . . . . . . 17 |- ( ( ( f e. Q. /\ g e. Q. ) /\ C e. P. ) -> ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( f G t ) ) <-> E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
22		r19.41v 2633	. . . . . . . . . . . . . . . . 17 |- ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( f G t ) ) <-> ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) )
23	21, 22	bitr3di 195	. . . . . . . . . . . . . . . 16 |- ( ( ( f e. Q. /\ g e. Q. ) /\ C e. P. ) -> ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
24	23	an32s 568	. . . . . . . . . . . . . . 15 |- ( ( ( f e. Q. /\ C e. P. ) /\ g e. Q. ) -> ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
25	6, 24	sylan2 286	. . . . . . . . . . . . . 14 |- ( ( ( f e. Q. /\ C e. P. ) /\ ( B e. P. /\ g e. ( 2nd ` B ) ) ) -> ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
26	25	anassrs 400	. . . . . . . . . . . . 13 |- ( ( ( ( f e. Q. /\ C e. P. ) /\ B e. P. ) /\ g e. ( 2nd ` B ) ) -> ( E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
27	26	rexbidva 2474	. . . . . . . . . . . 12 |- ( ( ( f e. Q. /\ C e. P. ) /\ B e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> E. g e. ( 2nd ` B ) ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
28		r19.41v 2633	. . . . . . . . . . . 12 |- ( E. g e. ( 2nd ` B ) ( E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) <-> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) )
29	27, 28	bitrdi 196	. . . . . . . . . . 11 |- ( ( ( f e. Q. /\ C e. P. ) /\ B e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
30	29	an31s 570	. . . . . . . . . 10 |- ( ( ( B e. P. /\ C e. P. ) /\ f e. Q. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
31	30	exbidv 1825	. . . . . . . . 9 |- ( ( ( B e. P. /\ C e. P. ) /\ f e. Q. ) -> ( E. t E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> E. t ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
32		genpelvl.2	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
33	32	caovcl 6029	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
34		elisset 2752	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( g G h ) e. Q. -> E. t t = ( g G h ) )
35	33, 34	syl 14	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( g e. Q. /\ h e. Q. ) -> E. t t = ( g G h ) )
36	35	biantrurd 305	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( g e. Q. /\ h e. Q. ) -> ( x = ( ( f G g ) G h ) <-> ( E. t t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
37		19.41v 1902	. . . . . . . . . . . . . . . . . . . . 21 |- ( E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> ( E. t t = ( g G h ) /\ x = ( ( f G g ) G h ) ) )
38	36, 37	bitr4di 198	. . . . . . . . . . . . . . . . . . . 20 |- ( ( g e. Q. /\ h e. Q. ) -> ( x = ( ( f G g ) G h ) <-> E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
39	9, 38	sylan2 286	. . . . . . . . . . . . . . . . . . 19 |- ( ( g e. Q. /\ ( C e. P. /\ h e. ( 2nd ` C ) ) ) -> ( x = ( ( f G g ) G h ) <-> E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
40	39	anassrs 400	. . . . . . . . . . . . . . . . . 18 |- ( ( ( g e. Q. /\ C e. P. ) /\ h e. ( 2nd ` C ) ) -> ( x = ( ( f G g ) G h ) <-> E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
41	40	rexbidva 2474	. . . . . . . . . . . . . . . . 17 |- ( ( g e. Q. /\ C e. P. ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. h e. ( 2nd ` C ) E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
42		rexcom4 2761	. . . . . . . . . . . . . . . . 17 |- ( E. h e. ( 2nd ` C ) E. t ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) )
43	41, 42	bitrdi 196	. . . . . . . . . . . . . . . 16 |- ( ( g e. Q. /\ C e. P. ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
44	43	ancoms 268	. . . . . . . . . . . . . . 15 |- ( ( C e. P. /\ g e. Q. ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
45	6, 44	sylan2 286	. . . . . . . . . . . . . 14 |- ( ( C e. P. /\ ( B e. P. /\ g e. ( 2nd ` B ) ) ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
46	45	anassrs 400	. . . . . . . . . . . . 13 |- ( ( ( C e. P. /\ B e. P. ) /\ g e. ( 2nd ` B ) ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
47	46	rexbidva 2474	. . . . . . . . . . . 12 |- ( ( C e. P. /\ B e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. g e. ( 2nd ` B ) E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
48	47	ancoms 268	. . . . . . . . . . 11 |- ( ( B e. P. /\ C e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. g e. ( 2nd ` B ) E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
49		rexcom4 2761	. . . . . . . . . . 11 |- ( E. g e. ( 2nd ` B ) E. t E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) <-> E. t E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) )
50	48, 49	bitrdi 196	. . . . . . . . . 10 |- ( ( B e. P. /\ C e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
51	50	adantr 276	. . . . . . . . 9 |- ( ( ( B e. P. /\ C e. P. ) /\ f e. Q. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) ( t = ( g G h ) /\ x = ( ( f G g ) G h ) ) ) )
52		df-rex 2461	. . . . . . . . . . 11 |- ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. t ( t e. ( 2nd ` ( B F C ) ) /\ x = ( f G t ) ) )
53		genpelvl.1	. . . . . . . . . . . . . 14 |- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
54	53, 32	genpelvu 7512	. . . . . . . . . . . . 13 |- ( ( B e. P. /\ C e. P. ) -> ( t e. ( 2nd ` ( B F C ) ) <-> E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) ) )
55	54	anbi1d 465	. . . . . . . . . . . 12 |- ( ( B e. P. /\ C e. P. ) -> ( ( t e. ( 2nd ` ( B F C ) ) /\ x = ( f G t ) ) <-> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
56	55	exbidv 1825	. . . . . . . . . . 11 |- ( ( B e. P. /\ C e. P. ) -> ( E. t ( t e. ( 2nd ` ( B F C ) ) /\ x = ( f G t ) ) <-> E. t ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
57	52, 56	bitrid 192	. . . . . . . . . 10 |- ( ( B e. P. /\ C e. P. ) -> ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. t ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
58	57	adantr 276	. . . . . . . . 9 |- ( ( ( B e. P. /\ C e. P. ) /\ f e. Q. ) -> ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. t ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) t = ( g G h ) /\ x = ( f G t ) ) ) )
59	31, 51, 58	3bitr4rd 221	. . . . . . . 8 |- ( ( ( B e. P. /\ C e. P. ) /\ f e. Q. ) -> ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
60	3, 59	sylan2 286	. . . . . . 7 |- ( ( ( B e. P. /\ C e. P. ) /\ ( A e. P. /\ f e. ( 2nd ` A ) ) ) -> ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
61	60	anassrs 400	. . . . . 6 |- ( ( ( ( B e. P. /\ C e. P. ) /\ A e. P. ) /\ f e. ( 2nd ` A ) ) -> ( E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
62	61	rexbidva 2474	. . . . 5 |- ( ( ( B e. P. /\ C e. P. ) /\ A e. P. ) -> ( E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
63	62	ancoms 268	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
64	63	3impb 1199	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
65		genpassg.5	. . . . . 6 |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) e. P. )
66	65	caovcl 6029	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
67	53, 32	genpelvu 7512	. . . . 5 |- ( ( A e. P. /\ ( B F C ) e. P. ) -> ( x e. ( 2nd ` ( A F ( B F C ) ) ) <-> E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) ) )
68	66, 67	sylan2 286	. . . 4 |- ( ( A e. P. /\ ( B e. P. /\ C e. P. ) ) -> ( x e. ( 2nd ` ( A F ( B F C ) ) ) <-> E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) ) )
69	68	3impb 1199	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` ( A F ( B F C ) ) ) <-> E. f e. ( 2nd ` A ) E. t e. ( 2nd ` ( B F C ) ) x = ( f G t ) ) )
70		df-rex 2461	. . . . 5 |- ( E. t e. ( 2nd ` ( A F B ) ) E. h e. ( 2nd ` C ) x = ( t G h ) <-> E. t ( t e. ( 2nd ` ( A F B ) ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
71	53, 32	genpelvu 7512	. . . . . . . 8 |- ( ( A e. P. /\ B e. P. ) -> ( t e. ( 2nd ` ( A F B ) ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) ) )
72	71	3adant3 1017	. . . . . . 7 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( t e. ( 2nd ` ( A F B ) ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) ) )
73	72	anbi1d 465	. . . . . 6 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( ( t e. ( 2nd ` ( A F B ) ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
74	73	exbidv 1825	. . . . 5 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. t ( t e. ( 2nd ` ( A F B ) ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
75	70, 74	bitrid 192	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. t e. ( 2nd ` ( A F B ) ) E. h e. ( 2nd ` C ) x = ( t G h ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
76	65	caovcl 6029	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
77	53, 32	genpelvu 7512	. . . . . 6 |- ( ( ( A F B ) e. P. /\ C e. P. ) -> ( x e. ( 2nd ` ( ( A F B ) F C ) ) <-> E. t e. ( 2nd ` ( A F B ) ) E. h e. ( 2nd ` C ) x = ( t G h ) ) )
78	76, 77	sylan 283	. . . . 5 |- ( ( ( A e. P. /\ B e. P. ) /\ C e. P. ) -> ( x e. ( 2nd ` ( ( A F B ) F C ) ) <-> E. t e. ( 2nd ` ( A F B ) ) E. h e. ( 2nd ` C ) x = ( t G h ) ) )
79	78	3impa 1194	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` ( ( A F B ) F C ) ) <-> E. t e. ( 2nd ` ( A F B ) ) E. h e. ( 2nd ` C ) x = ( t G h ) ) )
80	32	caovcl 6029	. . . . . . . . . . . . . . . . . . 19 |- ( ( f e. Q. /\ g e. Q. ) -> ( f G g ) e. Q. )
81		elisset 2752	. . . . . . . . . . . . . . . . . . 19 |- ( ( f G g ) e. Q. -> E. t t = ( f G g ) )
82	80, 81	syl 14	. . . . . . . . . . . . . . . . . 18 |- ( ( f e. Q. /\ g e. Q. ) -> E. t t = ( f G g ) )
83	82	biantrurd 305	. . . . . . . . . . . . . . . . 17 |- ( ( f e. Q. /\ g e. Q. ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> ( E. t t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) ) )
84		oveq1 5882	. . . . . . . . . . . . . . . . . . . . . 22 |- ( t = ( f G g ) -> ( t G h ) = ( ( f G g ) G h ) )
85	84	eqeq2d 2189	. . . . . . . . . . . . . . . . . . . . 21 |- ( t = ( f G g ) -> ( x = ( t G h ) <-> x = ( ( f G g ) G h ) ) )
86	85	rexbidv 2478	. . . . . . . . . . . . . . . . . . . 20 |- ( t = ( f G g ) -> ( E. h e. ( 2nd ` C ) x = ( t G h ) <-> E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
87	86	pm5.32i 454	. . . . . . . . . . . . . . . . . . 19 |- ( ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
88	87	exbii 1605	. . . . . . . . . . . . . . . . . 18 |- ( E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
89		19.41v 1902	. . . . . . . . . . . . . . . . . 18 |- ( E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) <-> ( E. t t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
90	88, 89	bitri 184	. . . . . . . . . . . . . . . . 17 |- ( E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( E. t t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
91	83, 90	bitr4di 198	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
92	6, 91	sylan2 286	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ ( B e. P. /\ g e. ( 2nd ` B ) ) ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
93	92	anassrs 400	. . . . . . . . . . . . . 14 |- ( ( ( f e. Q. /\ B e. P. ) /\ g e. ( 2nd ` B ) ) -> ( E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
94	93	rexbidva 2474	. . . . . . . . . . . . 13 |- ( ( f e. Q. /\ B e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. g e. ( 2nd ` B ) E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
95		rexcom4 2761	. . . . . . . . . . . . 13 |- ( E. g e. ( 2nd ` B ) E. t ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
96	94, 95	bitrdi 196	. . . . . . . . . . . 12 |- ( ( f e. Q. /\ B e. P. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
97	96	ancoms 268	. . . . . . . . . . 11 |- ( ( B e. P. /\ f e. Q. ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
98	3, 97	sylan2 286	. . . . . . . . . 10 |- ( ( B e. P. /\ ( A e. P. /\ f e. ( 2nd ` A ) ) ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
99	98	anassrs 400	. . . . . . . . 9 |- ( ( ( B e. P. /\ A e. P. ) /\ f e. ( 2nd ` A ) ) -> ( E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
100	99	rexbidva 2474	. . . . . . . 8 |- ( ( B e. P. /\ A e. P. ) -> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. f e. ( 2nd ` A ) E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
101		rexcom4 2761	. . . . . . . 8 |- ( E. f e. ( 2nd ` A ) E. t E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. t E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
102	100, 101	bitrdi 196	. . . . . . 7 |- ( ( B e. P. /\ A e. P. ) -> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
103		r19.41v 2633	. . . . . . . . . 10 |- ( E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
104	103	rexbii 2484	. . . . . . . . 9 |- ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. f e. ( 2nd ` A ) ( E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
105		r19.41v 2633	. . . . . . . . 9 |- ( E. f e. ( 2nd ` A ) ( E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
106	104, 105	bitri 184	. . . . . . . 8 |- ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
107	106	exbii 1605	. . . . . . 7 |- ( E. t E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) ( t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) )
108	102, 107	bitrdi 196	. . . . . 6 |- ( ( B e. P. /\ A e. P. ) -> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
109	108	ancoms 268	. . . . 5 |- ( ( A e. P. /\ B e. P. ) -> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
110	109	3adant3 1017	. . . 4 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) <-> E. t ( E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) t = ( f G g ) /\ E. h e. ( 2nd ` C ) x = ( t G h ) ) ) )
111	75, 79, 110	3bitr4d 220	. . 3 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` ( ( A F B ) F C ) ) <-> E. f e. ( 2nd ` A ) E. g e. ( 2nd ` B ) E. h e. ( 2nd ` C ) x = ( ( f G g ) G h ) ) )
112	64, 69, 111	3bitr4rd 221	. 2 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( x e. ( 2nd ` ( ( A F B ) F C ) ) <-> x e. ( 2nd ` ( A F ( B F C ) ) ) ) )
113	112	eqrdv 2175	1 |- ( ( A e. P. /\ B e. P. /\ C e. P. ) -> ( 2nd ` ( ( A F B ) F C ) ) = ( 2nd ` ( A F ( B F C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   {crab 2459   <.cop 3596   X. cxp 4625   dom cdm 4627   ` cfv 5217  (class class class)co 5875   e. cmpo 5877   1stc1st 6139   2ndc2nd 6140   Q.cnq 7279   P.cnp 7290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-qs 6541  df-ni 7303  df-nqqs 7347  df-inp 7465

This theorem is referenced by:  genpassg  7525