Theorem pi1xfrcnv 22857

Description: Given a path F between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
pi1xfr.p	|- P = ( J pi1 ( F ` 0 ) )
pi1xfr.q	|- Q = ( J pi1 ( F ` 1 ) )
pi1xfr.b	|- B = ( Base ` P )
pi1xfr.g	|- G = ran ( g e. U. B |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. )
pi1xfr.j	|- ( ph -> J e. (TopOn ` X ) )
pi1xfr.f	|- ( ph -> F e. ( II Cn J ) )
pi1xfr.i	|- I = ( x e. ( 0 [,] 1 ) |-> ( F ` ( 1 - x ) ) )
pi1xfrcnv.h	|- H = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. )

Assertion

Ref	Expression
pi1xfrcnv	|- ( ph -> ( `' G = H /\ `' G e. ( Q GrpHom P ) ) )

Distinct variable groups:   g, h, x, B   g, F, h, x   g, I, h, x   h, G   ph, g, h, x   g, J, h, x   P, g, h, x   Q, g, h, x

Allowed substitution hints:   G( x, g)   H( x, g, h)   X( x, g, h)

Proof of Theorem pi1xfrcnv

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pi1xfr.p	. . . 4 |- P = ( J pi1 ( F ` 0 ) )
2		pi1xfr.q	. . . 4 |- Q = ( J pi1 ( F ` 1 ) )
3		pi1xfr.b	. . . 4 |- B = ( Base ` P )
4		pi1xfr.g	. . . 4 |- G = ran ( g e. U. B |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. )
5		pi1xfr.j	. . . 4 |- ( ph -> J e. (TopOn ` X ) )
6		pi1xfr.f	. . . 4 |- ( ph -> F e. ( II Cn J ) )
7		pi1xfr.i	. . . 4 |- I = ( x e. ( 0 [,] 1 ) |-> ( F ` ( 1 - x ) ) )
8		pi1xfrcnv.h	. . . 4 |- H = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. )
9	1, 2, 3, 4, 5, 6, 7, 8	pi1xfrcnvlem 22856	. . 3 |- ( ph -> `' G C_ H )
10		fvex 6201	. . . . . . . 8 |- ( ~=ph ` J ) e. _V
11		ecexg 7746	. . . . . . . 8 |- ( ( ~=ph ` J ) e. _V -> [ h ] ( ~=ph ` J ) e. _V )
12	10, 11	mp1i 13	. . . . . . 7 |- ( ( ph /\ h e. U. ( Base ` Q ) ) -> [ h ] ( ~=ph ` J ) e. _V )
13		ecexg 7746	. . . . . . . 8 |- ( ( ~=ph ` J ) e. _V -> [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) e. _V )
14	10, 13	mp1i 13	. . . . . . 7 |- ( ( ph /\ h e. U. ( Base ` Q ) ) -> [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) e. _V )
15	8, 12, 14	fliftrel 6558	. . . . . 6 |- ( ph -> H C_ ( _V X. _V ) )
16		df-rel 5121	. . . . . 6 |- ( Rel H <-> H C_ ( _V X. _V ) )
17	15, 16	sylibr 224	. . . . 5 |- ( ph -> Rel H )
18		dfrel2 5583	. . . . 5 |- ( Rel H <-> `' `' H = H )
19	17, 18	sylib 208	. . . 4 |- ( ph -> `' `' H = H )
20		0elunit 12290	. . . . . . . . . 10 |- 0 e. ( 0 [,] 1 )
21		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = 0 -> ( 1 - x ) = ( 1 - 0 ) )
22		1m0e1 11131	. . . . . . . . . . . . 13 |- ( 1 - 0 ) = 1
23	21, 22	syl6eq 2672	. . . . . . . . . . . 12 |- ( x = 0 -> ( 1 - x ) = 1 )
24	23	fveq2d 6195	. . . . . . . . . . 11 |- ( x = 0 -> ( F ` ( 1 - x ) ) = ( F ` 1 ) )
25		fvex 6201	. . . . . . . . . . 11 |- ( F ` 1 ) e. _V
26	24, 7, 25	fvmpt 6282	. . . . . . . . . 10 |- ( 0 e. ( 0 [,] 1 ) -> ( I ` 0 ) = ( F ` 1 ) )
27	20, 26	ax-mp 5	. . . . . . . . 9 |- ( I ` 0 ) = ( F ` 1 )
28	27	oveq2i 6661	. . . . . . . 8 |- ( J pi1 ( I ` 0 ) ) = ( J pi1 ( F ` 1 ) )
29	2, 28	eqtr4i 2647	. . . . . . 7 |- Q = ( J pi1 ( I ` 0 ) )
30		1elunit 12291	. . . . . . . . . 10 |- 1 e. ( 0 [,] 1 )
31		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = 1 -> ( 1 - x ) = ( 1 - 1 ) )
32	31	fveq2d 6195	. . . . . . . . . . . 12 |- ( x = 1 -> ( F ` ( 1 - x ) ) = ( F ` ( 1 - 1 ) ) )
33		1m1e0 11089	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
34	33	fveq2i 6194	. . . . . . . . . . . 12 |- ( F ` ( 1 - 1 ) ) = ( F ` 0 )
35	32, 34	syl6eq 2672	. . . . . . . . . . 11 |- ( x = 1 -> ( F ` ( 1 - x ) ) = ( F ` 0 ) )
36		fvex 6201	. . . . . . . . . . 11 |- ( F ` 0 ) e. _V
37	35, 7, 36	fvmpt 6282	. . . . . . . . . 10 |- ( 1 e. ( 0 [,] 1 ) -> ( I ` 1 ) = ( F ` 0 ) )
38	30, 37	ax-mp 5	. . . . . . . . 9 |- ( I ` 1 ) = ( F ` 0 )
39	38	oveq2i 6661	. . . . . . . 8 |- ( J pi1 ( I ` 1 ) ) = ( J pi1 ( F ` 0 ) )
40	1, 39	eqtr4i 2647	. . . . . . 7 |- P = ( J pi1 ( I ` 1 ) )
41		eqid 2622	. . . . . . 7 |- ( Base ` Q ) = ( Base ` Q )
42		eqid 2622	. . . . . . 7 |- ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. )
43	7	pcorevcl 22825	. . . . . . . . 9 |- ( F e. ( II Cn J ) -> ( I e. ( II Cn J ) /\ ( I ` 0 ) = ( F ` 1 ) /\ ( I ` 1 ) = ( F ` 0 ) ) )
44	6, 43	syl 17	. . . . . . . 8 |- ( ph -> ( I e. ( II Cn J ) /\ ( I ` 0 ) = ( F ` 1 ) /\ ( I ` 1 ) = ( F ` 0 ) ) )
45	44	simp1d 1073	. . . . . . 7 |- ( ph -> I e. ( II Cn J ) )
46		oveq2 6658	. . . . . . . . 9 |- ( z = y -> ( 1 - z ) = ( 1 - y ) )
47	46	fveq2d 6195	. . . . . . . 8 |- ( z = y -> ( I ` ( 1 - z ) ) = ( I ` ( 1 - y ) ) )
48	47	cbvmptv 4750	. . . . . . 7 |- ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( I ` ( 1 - y ) ) )
49		eqid 2622	. . . . . . 7 |- ran ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. ) = ran ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. )
50	29, 40, 41, 42, 5, 45, 48, 49	pi1xfrcnvlem 22856	. . . . . 6 |- ( ph -> `' ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) C_ ran ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. ) )
51		iitopon 22682	. . . . . . . . . . . . . . . . 17 |- II e. (TopOn ` ( 0 [,] 1 ) )
52	51	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
53		cnf2 21053	. . . . . . . . . . . . . . . 16 |- ( ( II e. (TopOn ` ( 0 [,] 1 ) ) /\ J e. (TopOn ` X ) /\ F e. ( II Cn J ) ) -> F : ( 0 [,] 1 ) --> X )
54	52, 5, 6, 53	syl3anc 1326	. . . . . . . . . . . . . . 15 |- ( ph -> F : ( 0 [,] 1 ) --> X )
55	54	feqmptd 6249	. . . . . . . . . . . . . 14 |- ( ph -> F = ( z e. ( 0 [,] 1 ) |-> ( F ` z ) ) )
56		iirev 22728	. . . . . . . . . . . . . . . . 17 |- ( z e. ( 0 [,] 1 ) -> ( 1 - z ) e. ( 0 [,] 1 ) )
57		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 |- ( x = ( 1 - z ) -> ( 1 - x ) = ( 1 - ( 1 - z ) ) )
58	57	fveq2d 6195	. . . . . . . . . . . . . . . . . 18 |- ( x = ( 1 - z ) -> ( F ` ( 1 - x ) ) = ( F ` ( 1 - ( 1 - z ) ) ) )
59		fvex 6201	. . . . . . . . . . . . . . . . . 18 |- ( F ` ( 1 - ( 1 - z ) ) ) e. _V
60	58, 7, 59	fvmpt 6282	. . . . . . . . . . . . . . . . 17 |- ( ( 1 - z ) e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` ( 1 - ( 1 - z ) ) ) )
61	56, 60	syl 17	. . . . . . . . . . . . . . . 16 |- ( z e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` ( 1 - ( 1 - z ) ) ) )
62		ax-1cn 9994	. . . . . . . . . . . . . . . . . 18 |- 1 e. CC
63		unitssre 12319	. . . . . . . . . . . . . . . . . . . 20 |- ( 0 [,] 1 ) C_ RR
64	63	sseli 3599	. . . . . . . . . . . . . . . . . . 19 |- ( z e. ( 0 [,] 1 ) -> z e. RR )
65	64	recnd 10068	. . . . . . . . . . . . . . . . . 18 |- ( z e. ( 0 [,] 1 ) -> z e. CC )
66		nncan 10310	. . . . . . . . . . . . . . . . . 18 |- ( ( 1 e. CC /\ z e. CC ) -> ( 1 - ( 1 - z ) ) = z )
67	62, 65, 66	sylancr 695	. . . . . . . . . . . . . . . . 17 |- ( z e. ( 0 [,] 1 ) -> ( 1 - ( 1 - z ) ) = z )
68	67	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( z e. ( 0 [,] 1 ) -> ( F ` ( 1 - ( 1 - z ) ) ) = ( F ` z ) )
69	61, 68	eqtrd 2656	. . . . . . . . . . . . . . 15 |- ( z e. ( 0 [,] 1 ) -> ( I ` ( 1 - z ) ) = ( F ` z ) )
70	69	mpteq2ia 4740	. . . . . . . . . . . . . 14 |- ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) = ( z e. ( 0 [,] 1 ) |-> ( F ` z ) )
71	55, 70	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ph -> F = ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) )
72	71	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( F ( *p ` J ) ( h ( *p ` J ) I ) ) = ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) )
73	72	eceq1d 7783	. . . . . . . . . . 11 |- ( ph -> [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) = [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) )
74	73	opeq2d 4409	. . . . . . . . . 10 |- ( ph -> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. = <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. )
75	74	mpteq2dv 4745	. . . . . . . . 9 |- ( ph -> ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) = ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) )
76	75	rneqd 5353	. . . . . . . 8 |- ( ph -> ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( F ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) )
77	8, 76	syl5eq 2668	. . . . . . 7 |- ( ph -> H = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) )
78	77	cnveqd 5298	. . . . . 6 |- ( ph -> `' H = `' ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) )
79	3	a1i 11	. . . . . . . . . 10 |- ( ph -> B = ( Base ` P ) )
80	79	unieqd 4446	. . . . . . . . 9 |- ( ph -> U. B = U. ( Base ` P ) )
81	71	oveq2d 6666	. . . . . . . . . . . 12 |- ( ph -> ( g ( *p ` J ) F ) = ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) )
82	81	oveq2d 6666	. . . . . . . . . . 11 |- ( ph -> ( I ( *p ` J ) ( g ( *p ` J ) F ) ) = ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) )
83	82	eceq1d 7783	. . . . . . . . . 10 |- ( ph -> [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) = [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) )
84	83	opeq2d 4409	. . . . . . . . 9 |- ( ph -> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. = <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. )
85	80, 84	mpteq12dv 4733	. . . . . . . 8 |- ( ph -> ( g e. U. B |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. ) = ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. ) )
86	85	rneqd 5353	. . . . . . 7 |- ( ph -> ran ( g e. U. B |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) F ) ) ] ( ~=ph ` J ) >. ) = ran ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. ) )
87	4, 86	syl5eq 2668	. . . . . 6 |- ( ph -> G = ran ( g e. U. ( Base ` P ) |-> <. [ g ] ( ~=ph ` J ) , [ ( I ( *p ` J ) ( g ( *p ` J ) ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ) ) ] ( ~=ph ` J ) >. ) )
88	50, 78, 87	3sstr4d 3648	. . . . 5 |- ( ph -> `' H C_ G )
89		cnvss 5294	. . . . 5 |- ( `' H C_ G -> `' `' H C_ `' G )
90	88, 89	syl 17	. . . 4 |- ( ph -> `' `' H C_ `' G )
91	19, 90	eqsstr3d 3640	. . 3 |- ( ph -> H C_ `' G )
92	9, 91	eqssd 3620	. 2 |- ( ph -> `' G = H )
93	92, 77	eqtrd 2656	. . 3 |- ( ph -> `' G = ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) )
94	29, 40, 41, 42, 5, 45, 48	pi1xfr 22855	. . 3 |- ( ph -> ran ( h e. U. ( Base ` Q ) |-> <. [ h ] ( ~=ph ` J ) , [ ( ( z e. ( 0 [,] 1 ) |-> ( I ` ( 1 - z ) ) ) ( *p ` J ) ( h ( *p ` J ) I ) ) ] ( ~=ph ` J ) >. ) e. ( Q GrpHom P ) )
95	93, 94	eqeltrd 2701	. 2 |- ( ph -> `' G e. ( Q GrpHom P ) )
96	92, 95	jca 554	1 |- ( ph -> ( `' G = H /\ `' G e. ( Q GrpHom P ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   U.cuni 4436   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   [cec 7740   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   - cmin 10266   [,]cicc 12178   Basecbs 15857   GrpHom cghm 17657  TopOnctopon 20715   Cn ccn 21028   IIcii 22678   ~=ph cphtpc 22768   *pcpco 22800   pi1 cpi1 22803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-qus 16169  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-mulg 17541  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127  df-ii 22680  df-htpy 22769  df-phtpy 22770  df-phtpc 22791  df-pco 22805  df-om1 22806  df-pi1 22808

This theorem is referenced by:  pi1xfrgim  22858