Theorem htpycc 22779

Description: Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
htpycc.1	|- N = ( x e. X , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) )
htpycc.2	|- ( ph -> J e. (TopOn ` X ) )
htpycc.4	|- ( ph -> F e. ( J Cn K ) )
htpycc.5	|- ( ph -> G e. ( J Cn K ) )
htpycc.6	|- ( ph -> H e. ( J Cn K ) )
htpycc.7	|- ( ph -> L e. ( F ( J Htpy K ) G ) )
htpycc.8	|- ( ph -> M e. ( G ( J Htpy K ) H ) )

Assertion

Ref	Expression
htpycc	|- ( ph -> N e. ( F ( J Htpy K ) H ) )

Distinct variable groups:   x, y, J   x, K, y   x, L, y   x, M, y   x, X, y   ph, x, y

Allowed substitution hints:   F( x, y)   G( x, y)   H( x, y)   N( x, y)

Proof of Theorem htpycc

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

Step	Hyp	Ref	Expression
1		htpycc.2	. 2 |- ( ph -> J e. (TopOn ` X ) )
2		htpycc.4	. 2 |- ( ph -> F e. ( J Cn K ) )
3		htpycc.6	. 2 |- ( ph -> H e. ( J Cn K ) )
4		htpycc.1	. . 3 |- N = ( x e. X , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) )
5		iitopon 22682	. . . . 5 |- II e. (TopOn ` ( 0 [,] 1 ) )
6	5	a1i 11	. . . 4 |- ( ph -> II e. (TopOn ` ( 0 [,] 1 ) ) )
7		eqid 2622	. . . . 5 |- ( topGen ` ran (,) ) = ( topGen ` ran (,) )
8		eqid 2622	. . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
9		eqid 2622	. . . . 5 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
10		dfii2 22685	. . . . 5 |- II = ( ( topGen ` ran (,) ) ↾t ( 0 [,] 1 ) )
11		0red 10041	. . . . 5 |- ( ph -> 0 e. RR )
12		1red 10055	. . . . 5 |- ( ph -> 1 e. RR )
13		halfre 11246	. . . . . . 7 |- ( 1 / 2 ) e. RR
14		0re 10040	. . . . . . . 8 |- 0 e. RR
15		halfgt0 11248	. . . . . . . 8 |- 0 < ( 1 / 2 )
16	14, 13, 15	ltleii 10160	. . . . . . 7 |- 0 <_ ( 1 / 2 )
17		1re 10039	. . . . . . . 8 |- 1 e. RR
18		halflt1 11250	. . . . . . . 8 |- ( 1 / 2 ) < 1
19	13, 17, 18	ltleii 10160	. . . . . . 7 |- ( 1 / 2 ) <_ 1
20	14, 17	elicc2i 12239	. . . . . . 7 |- ( ( 1 / 2 ) e. ( 0 [,] 1 ) <-> ( ( 1 / 2 ) e. RR /\ 0 <_ ( 1 / 2 ) /\ ( 1 / 2 ) <_ 1 ) )
21	13, 16, 19, 20	mpbir3an 1244	. . . . . 6 |- ( 1 / 2 ) e. ( 0 [,] 1 )
22	21	a1i 11	. . . . 5 |- ( ph -> ( 1 / 2 ) e. ( 0 [,] 1 ) )
23		htpycc.5	. . . . . . . . . . . 12 |- ( ph -> G e. ( J Cn K ) )
24		htpycc.7	. . . . . . . . . . . 12 |- ( ph -> L e. ( F ( J Htpy K ) G ) )
25	1, 2, 23, 24	htpyi 22773	. . . . . . . . . . 11 |- ( ( ph /\ s e. X ) -> ( ( s L 0 ) = ( F ` s ) /\ ( s L 1 ) = ( G ` s ) ) )
26	25	simprd 479	. . . . . . . . . 10 |- ( ( ph /\ s e. X ) -> ( s L 1 ) = ( G ` s ) )
27		htpycc.8	. . . . . . . . . . . 12 |- ( ph -> M e. ( G ( J Htpy K ) H ) )
28	1, 23, 3, 27	htpyi 22773	. . . . . . . . . . 11 |- ( ( ph /\ s e. X ) -> ( ( s M 0 ) = ( G ` s ) /\ ( s M 1 ) = ( H ` s ) ) )
29	28	simpld 475	. . . . . . . . . 10 |- ( ( ph /\ s e. X ) -> ( s M 0 ) = ( G ` s ) )
30	26, 29	eqtr4d 2659	. . . . . . . . 9 |- ( ( ph /\ s e. X ) -> ( s L 1 ) = ( s M 0 ) )
31	30	ralrimiva 2966	. . . . . . . 8 |- ( ph -> A. s e. X ( s L 1 ) = ( s M 0 ) )
32		oveq1 6657	. . . . . . . . . 10 |- ( s = x -> ( s L 1 ) = ( x L 1 ) )
33		oveq1 6657	. . . . . . . . . 10 |- ( s = x -> ( s M 0 ) = ( x M 0 ) )
34	32, 33	eqeq12d 2637	. . . . . . . . 9 |- ( s = x -> ( ( s L 1 ) = ( s M 0 ) <-> ( x L 1 ) = ( x M 0 ) ) )
35	34	rspccva 3308	. . . . . . . 8 |- ( ( A. s e. X ( s L 1 ) = ( s M 0 ) /\ x e. X ) -> ( x L 1 ) = ( x M 0 ) )
36	31, 35	sylan 488	. . . . . . 7 |- ( ( ph /\ x e. X ) -> ( x L 1 ) = ( x M 0 ) )
37	36	adantrl 752	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L 1 ) = ( x M 0 ) )
38		simprl 794	. . . . . . . . 9 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> y = ( 1 / 2 ) )
39	38	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( 2 x. y ) = ( 2 x. ( 1 / 2 ) ) )
40		2cn 11091	. . . . . . . . 9 |- 2 e. CC
41		2ne0 11113	. . . . . . . . 9 |- 2 =/= 0
42	40, 41	recidi 10756	. . . . . . . 8 |- ( 2 x. ( 1 / 2 ) ) = 1
43	39, 42	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( 2 x. y ) = 1 )
44	43	oveq2d 6666	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L ( 2 x. y ) ) = ( x L 1 ) )
45	43	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( ( 2 x. y ) - 1 ) = ( 1 - 1 ) )
46		1m1e0 11089	. . . . . . . 8 |- ( 1 - 1 ) = 0
47	45, 46	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( ( 2 x. y ) - 1 ) = 0 )
48	47	oveq2d 6666	. . . . . 6 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x M ( ( 2 x. y ) - 1 ) ) = ( x M 0 ) )
49	37, 44, 48	3eqtr4d 2666	. . . . 5 |- ( ( ph /\ ( y = ( 1 / 2 ) /\ x e. X ) ) -> ( x L ( 2 x. y ) ) = ( x M ( ( 2 x. y ) - 1 ) ) )
50		retopon 22567	. . . . . . . 8 |- ( topGen ` ran (,) ) e. (TopOn ` RR )
51		iccssre 12255	. . . . . . . . 9 |- ( ( 0 e. RR /\ ( 1 / 2 ) e. RR ) -> ( 0 [,] ( 1 / 2 ) ) C_ RR )
52	14, 13, 51	mp2an 708	. . . . . . . 8 |- ( 0 [,] ( 1 / 2 ) ) C_ RR
53		resttopon 20965	. . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( 0 [,] ( 1 / 2 ) ) C_ RR ) -> ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) ) )
54	50, 52, 53	mp2an 708	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) )
55	54	a1i 11	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) e. (TopOn ` ( 0 [,] ( 1 / 2 ) ) ) )
56	55, 1	cnmpt2nd 21472	. . . . . 6 |- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> x ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn J ) )
57	55, 1	cnmpt1st 21471	. . . . . . 7 |- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> y ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) )
58	8	iihalf1cn 22731	. . . . . . . 8 |- ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
59	58	a1i 11	. . . . . . 7 |- ( ph -> ( z e. ( 0 [,] ( 1 / 2 ) ) |-> ( 2 x. z ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) )
60		oveq2 6658	. . . . . . 7 |- ( z = y -> ( 2 x. z ) = ( 2 x. y ) )
61	55, 1, 57, 55, 59, 60	cnmpt21 21474	. . . . . 6 |- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> ( 2 x. y ) ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn II ) )
62	1, 2, 23	htpycn 22772	. . . . . . 7 |- ( ph -> ( F ( J Htpy K ) G ) C_ ( ( J tX II ) Cn K ) )
63	62, 24	sseldd 3604	. . . . . 6 |- ( ph -> L e. ( ( J tX II ) Cn K ) )
64	55, 1, 56, 61, 63	cnmpt22f 21478	. . . . 5 |- ( ph -> ( y e. ( 0 [,] ( 1 / 2 ) ) , x e. X |-> ( x L ( 2 x. y ) ) ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) tX J ) Cn K ) )
65		iccssre 12255	. . . . . . . . 9 |- ( ( ( 1 / 2 ) e. RR /\ 1 e. RR ) -> ( ( 1 / 2 ) [,] 1 ) C_ RR )
66	13, 17, 65	mp2an 708	. . . . . . . 8 |- ( ( 1 / 2 ) [,] 1 ) C_ RR
67		resttopon 20965	. . . . . . . 8 |- ( ( ( topGen ` ran (,) ) e. (TopOn ` RR ) /\ ( ( 1 / 2 ) [,] 1 ) C_ RR ) -> ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) ) )
68	50, 66, 67	mp2an 708	. . . . . . 7 |- ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) )
69	68	a1i 11	. . . . . 6 |- ( ph -> ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) e. (TopOn ` ( ( 1 / 2 ) [,] 1 ) ) )
70	69, 1	cnmpt2nd 21472	. . . . . 6 |- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> x ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn J ) )
71	69, 1	cnmpt1st 21471	. . . . . . 7 |- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> y ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ) )
72	9	iihalf2cn 22733	. . . . . . . 8 |- ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II )
73	72	a1i 11	. . . . . . 7 |- ( ph -> ( z e. ( ( 1 / 2 ) [,] 1 ) |-> ( ( 2 x. z ) - 1 ) ) e. ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) )
74	60	oveq1d 6665	. . . . . . 7 |- ( z = y -> ( ( 2 x. z ) - 1 ) = ( ( 2 x. y ) - 1 ) )
75	69, 1, 71, 69, 73, 74	cnmpt21 21474	. . . . . 6 |- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> ( ( 2 x. y ) - 1 ) ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn II ) )
76	1, 23, 3	htpycn 22772	. . . . . . 7 |- ( ph -> ( G ( J Htpy K ) H ) C_ ( ( J tX II ) Cn K ) )
77	76, 27	sseldd 3604	. . . . . 6 |- ( ph -> M e. ( ( J tX II ) Cn K ) )
78	69, 1, 70, 75, 77	cnmpt22f 21478	. . . . 5 |- ( ph -> ( y e. ( ( 1 / 2 ) [,] 1 ) , x e. X |-> ( x M ( ( 2 x. y ) - 1 ) ) ) e. ( ( ( ( topGen ` ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) tX J ) Cn K ) )
79	7, 8, 9, 10, 11, 12, 22, 1, 49, 64, 78	cnmpt2pc 22727	. . . 4 |- ( ph -> ( y e. ( 0 [,] 1 ) , x e. X |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) ) e. ( ( II tX J ) Cn K ) )
80	6, 1, 79	cnmptcom 21481	. . 3 |- ( ph -> ( x e. X , y e. ( 0 [,] 1 ) |-> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) ) e. ( ( J tX II ) Cn K ) )
81	4, 80	syl5eqel 2705	. 2 |- ( ph -> N e. ( ( J tX II ) Cn K ) )
82		simpr 477	. . . 4 |- ( ( ph /\ s e. X ) -> s e. X )
83		0elunit 12290	. . . 4 |- 0 e. ( 0 [,] 1 )
84		simpr 477	. . . . . . . 8 |- ( ( x = s /\ y = 0 ) -> y = 0 )
85	84, 16	syl6eqbr 4692	. . . . . . 7 |- ( ( x = s /\ y = 0 ) -> y <_ ( 1 / 2 ) )
86	85	iftrued 4094	. . . . . 6 |- ( ( x = s /\ y = 0 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( x L ( 2 x. y ) ) )
87		simpl 473	. . . . . . 7 |- ( ( x = s /\ y = 0 ) -> x = s )
88	84	oveq2d 6666	. . . . . . . 8 |- ( ( x = s /\ y = 0 ) -> ( 2 x. y ) = ( 2 x. 0 ) )
89		2t0e0 11183	. . . . . . . 8 |- ( 2 x. 0 ) = 0
90	88, 89	syl6eq 2672	. . . . . . 7 |- ( ( x = s /\ y = 0 ) -> ( 2 x. y ) = 0 )
91	87, 90	oveq12d 6668	. . . . . 6 |- ( ( x = s /\ y = 0 ) -> ( x L ( 2 x. y ) ) = ( s L 0 ) )
92	86, 91	eqtrd 2656	. . . . 5 |- ( ( x = s /\ y = 0 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( s L 0 ) )
93		ovex 6678	. . . . 5 |- ( s L 0 ) e. _V
94	92, 4, 93	ovmpt2a 6791	. . . 4 |- ( ( s e. X /\ 0 e. ( 0 [,] 1 ) ) -> ( s N 0 ) = ( s L 0 ) )
95	82, 83, 94	sylancl 694	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( s L 0 ) )
96	25	simpld 475	. . 3 |- ( ( ph /\ s e. X ) -> ( s L 0 ) = ( F ` s ) )
97	95, 96	eqtrd 2656	. 2 |- ( ( ph /\ s e. X ) -> ( s N 0 ) = ( F ` s ) )
98		1elunit 12291	. . . 4 |- 1 e. ( 0 [,] 1 )
99	13, 17	ltnlei 10158	. . . . . . . . 9 |- ( ( 1 / 2 ) < 1 <-> -. 1 <_ ( 1 / 2 ) )
100	18, 99	mpbi 220	. . . . . . . 8 |- -. 1 <_ ( 1 / 2 )
101		simpr 477	. . . . . . . . 9 |- ( ( x = s /\ y = 1 ) -> y = 1 )
102	101	breq1d 4663	. . . . . . . 8 |- ( ( x = s /\ y = 1 ) -> ( y <_ ( 1 / 2 ) <-> 1 <_ ( 1 / 2 ) ) )
103	100, 102	mtbiri 317	. . . . . . 7 |- ( ( x = s /\ y = 1 ) -> -. y <_ ( 1 / 2 ) )
104	103	iffalsed 4097	. . . . . 6 |- ( ( x = s /\ y = 1 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( x M ( ( 2 x. y ) - 1 ) ) )
105		simpl 473	. . . . . . 7 |- ( ( x = s /\ y = 1 ) -> x = s )
106	101	oveq2d 6666	. . . . . . . . . 10 |- ( ( x = s /\ y = 1 ) -> ( 2 x. y ) = ( 2 x. 1 ) )
107		2t1e2 11176	. . . . . . . . . 10 |- ( 2 x. 1 ) = 2
108	106, 107	syl6eq 2672	. . . . . . . . 9 |- ( ( x = s /\ y = 1 ) -> ( 2 x. y ) = 2 )
109	108	oveq1d 6665	. . . . . . . 8 |- ( ( x = s /\ y = 1 ) -> ( ( 2 x. y ) - 1 ) = ( 2 - 1 ) )
110		2m1e1 11135	. . . . . . . 8 |- ( 2 - 1 ) = 1
111	109, 110	syl6eq 2672	. . . . . . 7 |- ( ( x = s /\ y = 1 ) -> ( ( 2 x. y ) - 1 ) = 1 )
112	105, 111	oveq12d 6668	. . . . . 6 |- ( ( x = s /\ y = 1 ) -> ( x M ( ( 2 x. y ) - 1 ) ) = ( s M 1 ) )
113	104, 112	eqtrd 2656	. . . . 5 |- ( ( x = s /\ y = 1 ) -> if ( y <_ ( 1 / 2 ) , ( x L ( 2 x. y ) ) , ( x M ( ( 2 x. y ) - 1 ) ) ) = ( s M 1 ) )
114		ovex 6678	. . . . 5 |- ( s M 1 ) e. _V
115	113, 4, 114	ovmpt2a 6791	. . . 4 |- ( ( s e. X /\ 1 e. ( 0 [,] 1 ) ) -> ( s N 1 ) = ( s M 1 ) )
116	82, 98, 115	sylancl 694	. . 3 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( s M 1 ) )
117	28	simprd 479	. . 3 |- ( ( ph /\ s e. X ) -> ( s M 1 ) = ( H ` s ) )
118	116, 117	eqtrd 2656	. 2 |- ( ( ph /\ s e. X ) -> ( s N 1 ) = ( H ` s ) )
119	1, 2, 3, 81, 97, 118	ishtpyd 22774	1 |- ( ph -> N e. ( F ( J Htpy K ) H ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   2c2 11070   (,)cioo 12175   [,]cicc 12178   ↾_t crest 16081   topGenctg 16098  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   IIcii 22678   Htpy chtpy 22766

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-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-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-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  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

This theorem is referenced by:  phtpycc  22790