Theorem genpassu 6623

Description: Associativity of upper cuts. Lemma for genpassg 6624. (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 6573	. . . . . . . . 9 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
2		elprnqu 6580	. . . . . . . . 9 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
3	1, 2	sylan 267	. . . . . . . 8 |- ( ( A e. P. /\ f e. ( 2nd ` A ) ) -> f e. Q. )
4		prop 6573	. . . . . . . . . . . . . . . 16 |- ( B e. P. -> <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. )
5		elprnqu 6580	. . . . . . . . . . . . . . . 16 |- ( ( <. ( 1st ` B ) , ( 2nd ` B ) >. e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
6	4, 5	sylan 267	. . . . . . . . . . . . . . 15 |- ( ( B e. P. /\ g e. ( 2nd ` B ) ) -> g e. Q. )
7		r19.41v 2466	. . . . . . . . . . . . . . . . 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 ) ) )
8		prop 6573	. . . . . . . . . . . . . . . . . . . . 21 |- ( C e. P. -> <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. )
9		elprnqu 6580	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( <. ( 1st ` C ) , ( 2nd ` C ) >. e. P. /\ h e. ( 2nd ` C ) ) -> h e. Q. )
10	8, 9	sylan 267	. . . . . . . . . . . . . . . . . . . 20 |- ( ( C e. P. /\ h e. ( 2nd ` C ) ) -> h e. Q. )
11		oveq2 5520	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( t = ( g G h ) -> ( f G t ) = ( f G ( g G h ) ) )
12	11	adantr 261	. . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
14	13	adantl 262	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f G g ) G h ) = ( f G ( g G h ) ) )
15	12, 14	eqtr4d 2075	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f G t ) = ( ( f G g ) G h ) )
16	15	eqeq2d 2051	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( t = ( g G h ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( x = ( f G t ) <-> x = ( ( f G g ) G h ) ) )
17	16	expcom 109	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( t = ( g G h ) -> ( x = ( f G t ) <-> x = ( ( f G g ) G h ) ) ) )
18	17	pm5.32d 423	. . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
19	18	3expa 1104	. . . . . . . . . . . . . . . . . . . 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 ) ) ) )
20	10, 19	sylan2 270	. . . . . . . . . . . . . . . . . . 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 ) ) ) )
21	20	anassrs 380	. . . . . . . . . . . . . . . . . 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 ) ) ) )
22	21	rexbidva 2323	. . . . . . . . . . . . . . . . 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 ) ) ) )
23	7, 22	syl5rbbr 184	. . . . . . . . . . . . . . . 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 502	. . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 380	. . . . . . . . . . . . 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 2323	. . . . . . . . . . . 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 2466	. . . . . . . . . . . 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	syl6bb 185	. . . . . . . . . . 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 504	. . . . . . . . . 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 1706	. . . . . . . . 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 5655	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( g e. Q. /\ h e. Q. ) -> ( g G h ) e. Q. )
34		elisset 2568	. . . . . . . . . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . . . . . . . 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 1782	. . . . . . . . . . . . . . . . . . . . 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	syl6bbr 187	. . . . . . . . . . . . . . . . . . . 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	10, 38	sylan2 270	. . . . . . . . . . . . . . . . . . 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 380	. . . . . . . . . . . . . . . . . 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 2323	. . . . . . . . . . . . . . . . 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 2577	. . . . . . . . . . . . . . . . 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	syl6bb 185	. . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . 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 380	. . . . . . . . . . . . 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 2323	. . . . . . . . . . . 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 255	. . . . . . . . . . 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 2577	. . . . . . . . . . 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	syl6bb 185	. . . . . . . . . 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 261	. . . . . . . . 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 2312	. . . . . . . . . . 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 6611	. . . . . . . . . . . . 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 438	. . . . . . . . . . . 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 1706	. . . . . . . . . . 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	syl5bb 181	. . . . . . . . . 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 261	. . . . . . . . 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 210	. . . . . . . 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 270	. . . . . . 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 380	. . . . . 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 2323	. . . . 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 255	. . . 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 1100	. . 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 5655	. . . . 5 |- ( ( B e. P. /\ C e. P. ) -> ( B F C ) e. P. )
67	53, 32	genpelvu 6611	. . . . 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 270	. . . 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 1100	. . 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 2312	. . . . 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 6611	. . . . . . . 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 924	. . . . . . 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 438	. . . . . 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 1706	. . . . 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	syl5bb 181	. . . 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 5655	. . . . . 6 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) e. P. )
77	53, 32	genpelvu 6611	. . . . . 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 267	. . . . 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 1099	. . . 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 5655	. . . . . . . . . . . . . . . . . . 19 |- ( ( f e. Q. /\ g e. Q. ) -> ( f G g ) e. Q. )
81		elisset 2568	. . . . . . . . . . . . . . . . . . 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 289	. . . . . . . . . . . . . . . . 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 5519	. . . . . . . . . . . . . . . . . . . . . 22 |- ( t = ( f G g ) -> ( t G h ) = ( ( f G g ) G h ) )
85	84	eqeq2d 2051	. . . . . . . . . . . . . . . . . . . . 21 |- ( t = ( f G g ) -> ( x = ( t G h ) <-> x = ( ( f G g ) G h ) ) )
86	85	rexbidv 2327	. . . . . . . . . . . . . . . . . . . 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 427	. . . . . . . . . . . . . . . . . . 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 1496	. . . . . . . . . . . . . . . . . 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 1782	. . . . . . . . . . . . . . . . . 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 173	. . . . . . . . . . . . . . . . 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	syl6bbr 187	. . . . . . . . . . . . . . . 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 270	. . . . . . . . . . . . . . 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 380	. . . . . . . . . . . . . 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 2323	. . . . . . . . . . . . 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 2577	. . . . . . . . . . . . 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	syl6bb 185	. . . . . . . . . . . 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 255	. . . . . . . . . . 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 270	. . . . . . . . . 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 380	. . . . . . . . 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 2323	. . . . . . . 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 2577	. . . . . . . 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	syl6bb 185	. . . . . . 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 2466	. . . . . . . . . 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 2331	. . . . . . . . 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 2466	. . . . . . . . 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 173	. . . . . . . 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 1496	. . . . . . 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	syl6bb 185	. . . . . 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 255	. . . . 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 924	. . . 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 209	. . 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 210	. 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 2038	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 97   <-> wb 98   /\ w3a 885   = wceq 1243   E.wex 1381   e. wcel 1393   E.wrex 2307   {crab 2310   <.cop 3378   X. cxp 4343   dom cdm 4345   ` cfv 4902  (class class class)co 5512   |-> cmpt2 5514   1stc1st 5765   2ndc2nd 5766   Q.cnq 6378   P.cnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6402  df-nqqs 6446  df-inp 6564

This theorem is referenced by:  genpassg  6624