Theorem constr1trl 25397

Description: Construction of a trail from one given edge in a graph. (Contributed by Alexander van der Vekens, 3-Dec-2017.)

Hypotheses

Ref	Expression
1trl.f	|- F = { <. 0 , i >. }
1trl.p	|- P = { <. 0 , A >. , <. 1 , B >. }

Assertion

Ref	Expression
constr1trl	|- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> F ( V Trails E ) P )

Proof of Theorem constr1trl

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1trl.f	. . 3 |- F = { <. 0 , i >. }
2		c0ex 9655	. . . . . . 7 |- 0 e. _V
3		vex 3034	. . . . . . 7 |- i e. _V
4	2, 3	f1osn 5866	. . . . . 6 |- { <. 0 , i >. } : { 0 } -1-1-onto-> { i }
5		f1of1 5827	. . . . . 6 |- ( { <. 0 , i >. } : { 0 } -1-1-onto-> { i } -> { <. 0 , i >. } : { 0 } -1-1-> { i } )
6		fveq2 5879	. . . . . . . . . . . . . . . 16 |- ( F = { <. 0 , i >. } -> ( # ` F ) = ( # ` { <. 0 , i >. } ) )
7		opex 4664	. . . . . . . . . . . . . . . . 17 |- <. 0 , i >. e. _V
8		hashsng 12587	. . . . . . . . . . . . . . . . 17 |- ( <. 0 , i >. e. _V -> ( # ` { <. 0 , i >. } ) = 1 )
9	7, 8	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( # ` { <. 0 , i >. } ) = 1
10	6, 9	syl6eq 2521	. . . . . . . . . . . . . . 15 |- ( F = { <. 0 , i >. } -> ( # ` F ) = 1 )
11	1, 10	ax-mp 5	. . . . . . . . . . . . . 14 |- ( # ` F ) = 1
12	11	oveq2i 6319	. . . . . . . . . . . . 13 |- ( 0..^ ( # ` F ) ) = ( 0..^ 1 )
13		fzo01 12024	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
14	12, 13	eqtri 2493	. . . . . . . . . . . 12 |- ( 0..^ ( # ` F ) ) = { 0 }
15	14	eqcomi 2480	. . . . . . . . . . 11 |- { 0 } = ( 0..^ ( # ` F ) )
16		f1eq2 5788	. . . . . . . . . . 11 |- ( { 0 } = ( 0..^ ( # ` F ) ) -> ( { <. 0 , i >. } : { 0 } -1-1-> { i } <-> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> { i } ) )
17	15, 16	ax-mp 5	. . . . . . . . . 10 |- ( { <. 0 , i >. } : { 0 } -1-1-> { i } <-> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> { i } )
18	17	biimpi 199	. . . . . . . . 9 |- ( { <. 0 , i >. } : { 0 } -1-1-> { i } -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> { i } )
19	18	adantr 472	. . . . . . . 8 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> { i } )
20		prid1g 4069	. . . . . . . . . . . . . 14 |- ( A e. V -> A e. { A , B } )
21	20	adantr 472	. . . . . . . . . . . . 13 |- ( ( A e. V /\ B e. V ) -> A e. { A , B } )
22	21	3ad2ant2 1052	. . . . . . . . . . . 12 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> A e. { A , B } )
23	22	adantl 473	. . . . . . . . . . 11 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> A e. { A , B } )
24		eleq2 2538	. . . . . . . . . . . . 13 |- ( ( E ` i ) = { A , B } -> ( A e. ( E ` i ) <-> A e. { A , B } ) )
25	24	3ad2ant3 1053	. . . . . . . . . . . 12 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( A e. ( E ` i ) <-> A e. { A , B } ) )
26	25	adantl 473	. . . . . . . . . . 11 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> ( A e. ( E ` i ) <-> A e. { A , B } ) )
27	23, 26	mpbird 240	. . . . . . . . . 10 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> A e. ( E ` i ) )
28		elfvdm 5905	. . . . . . . . . 10 |- ( A e. ( E ` i ) -> i e. dom E )
29	27, 28	syl 17	. . . . . . . . 9 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> i e. dom E )
30	29	snssd 4108	. . . . . . . 8 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> { i } C_ dom E )
31		f1ss 5797	. . . . . . . 8 |- ( ( { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> { i } /\ { i } C_ dom E ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E )
32	19, 30, 31	syl2anc 673	. . . . . . 7 |- ( ( { <. 0 , i >. } : { 0 } -1-1-> { i } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E )
33	32	ex 441	. . . . . 6 |- ( { <. 0 , i >. } : { 0 } -1-1-> { i } -> ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E ) )
34	4, 5, 33	mp2b 10	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E )
35	34	adantl 473	. . . 4 |- ( ( F = { <. 0 , i >. } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E )
36		f1eq1 5787	. . . . 5 |- ( F = { <. 0 , i >. } -> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E <-> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E ) )
37	36	adantr 472	. . . 4 |- ( ( F = { <. 0 , i >. } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E <-> { <. 0 , i >. } : ( 0..^ ( # ` F ) ) -1-1-> dom E ) )
38	35, 37	mpbird 240	. . 3 |- ( ( F = { <. 0 , i >. } /\ ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom E )
39	1, 38	mpan 684	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> F : ( 0..^ ( # ` F ) ) -1-1-> dom E )
40		1trl.p	. . . 4 |- P = { <. 0 , A >. , <. 1 , B >. }
41		0z 10972	. . . . . . . . 9 |- 0 e. ZZ
42		1z 10991	. . . . . . . . 9 |- 1 e. ZZ
43	41, 42	pm3.2i 462	. . . . . . . 8 |- ( 0 e. ZZ /\ 1 e. ZZ )
44		0ne1 10699	. . . . . . . 8 |- 0 =/= 1
45		fprg 6089	. . . . . . . . 9 |- ( ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( A e. V /\ B e. V ) /\ 0 =/= 1 ) -> { <. 0 , A >. , <. 1 , B >. } : { 0 , 1 } --> { A , B } )
46		0p1e1 10743	. . . . . . . . . . . . 13 |- ( 0 + 1 ) = 1
47	46	eqcomi 2480	. . . . . . . . . . . 12 |- 1 = ( 0 + 1 )
48	47	oveq2i 6319	. . . . . . . . . . 11 |- ( 0 ... 1 ) = ( 0 ... ( 0 + 1 ) )
49		fzpr 11877	. . . . . . . . . . . 12 |- ( 0 e. ZZ -> ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) } )
50	41, 49	ax-mp 5	. . . . . . . . . . 11 |- ( 0 ... ( 0 + 1 ) ) = { 0 , ( 0 + 1 ) }
51	46	preq2i 4046	. . . . . . . . . . 11 |- { 0 , ( 0 + 1 ) } = { 0 , 1 }
52	48, 50, 51	3eqtri 2497	. . . . . . . . . 10 |- ( 0 ... 1 ) = { 0 , 1 }
53	52	feq2i 5731	. . . . . . . . 9 |- ( { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> { A , B } <-> { <. 0 , A >. , <. 1 , B >. } : { 0 , 1 } --> { A , B } )
54	45, 53	sylibr 217	. . . . . . . 8 |- ( ( ( 0 e. ZZ /\ 1 e. ZZ ) /\ ( A e. V /\ B e. V ) /\ 0 =/= 1 ) -> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> { A , B } )
55	43, 44, 54	mp3an13 1381	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> { A , B } )
56		prssi 4119	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> { A , B } C_ V )
57	55, 56	fssd 5750	. . . . . 6 |- ( ( A e. V /\ B e. V ) -> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> V )
58	57	adantl 473	. . . . 5 |- ( ( P = { <. 0 , A >. , <. 1 , B >. } /\ ( A e. V /\ B e. V ) ) -> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> V )
59		id 22	. . . . . . 7 |- ( P = { <. 0 , A >. , <. 1 , B >. } -> P = { <. 0 , A >. , <. 1 , B >. } )
60	11	oveq2i 6319	. . . . . . . 8 |- ( 0 ... ( # ` F ) ) = ( 0 ... 1 )
61	60	a1i 11	. . . . . . 7 |- ( P = { <. 0 , A >. , <. 1 , B >. } -> ( 0 ... ( # ` F ) ) = ( 0 ... 1 ) )
62	59, 61	feq12d 5727	. . . . . 6 |- ( P = { <. 0 , A >. , <. 1 , B >. } -> ( P : ( 0 ... ( # ` F ) ) --> V <-> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> V ) )
63	62	adantr 472	. . . . 5 |- ( ( P = { <. 0 , A >. , <. 1 , B >. } /\ ( A e. V /\ B e. V ) ) -> ( P : ( 0 ... ( # ` F ) ) --> V <-> { <. 0 , A >. , <. 1 , B >. } : ( 0 ... 1 ) --> V ) )
64	58, 63	mpbird 240	. . . 4 |- ( ( P = { <. 0 , A >. , <. 1 , B >. } /\ ( A e. V /\ B e. V ) ) -> P : ( 0 ... ( # ` F ) ) --> V )
65	40, 64	mpan 684	. . 3 |- ( ( A e. V /\ B e. V ) -> P : ( 0 ... ( # ` F ) ) --> V )
66	65	3ad2ant2 1052	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> P : ( 0 ... ( # ` F ) ) --> V )
67		id 22	. . . . . 6 |- ( ( E ` i ) = { A , B } -> ( E ` i ) = { A , B } )
68	67	3ad2ant3 1053	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( E ` i ) = { A , B } )
69		fveq1 5878	. . . . . . . . 9 |- ( F = { <. 0 , i >. } -> ( F ` 0 ) = ( { <. 0 , i >. } ` 0 ) )
70	2, 3	fvsn 6113	. . . . . . . . 9 |- ( { <. 0 , i >. } ` 0 ) = i
71	69, 70	syl6eq 2521	. . . . . . . 8 |- ( F = { <. 0 , i >. } -> ( F ` 0 ) = i )
72	1, 71	ax-mp 5	. . . . . . 7 |- ( F ` 0 ) = i
73	72	a1i 11	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( F ` 0 ) = i )
74	73	fveq2d 5883	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( E ` ( F ` 0 ) ) = ( E ` i ) )
75		fveq1 5878	. . . . . . . . . 10 |- ( P = { <. 0 , A >. , <. 1 , B >. } -> ( P ` 0 ) = ( { <. 0 , A >. , <. 1 , B >. } ` 0 ) )
76		fvpr1g 6125	. . . . . . . . . . 11 |- ( ( 0 e. _V /\ A e. V /\ 0 =/= 1 ) -> ( { <. 0 , A >. , <. 1 , B >. } ` 0 ) = A )
77	2, 44, 76	mp3an13 1381	. . . . . . . . . 10 |- ( A e. V -> ( { <. 0 , A >. , <. 1 , B >. } ` 0 ) = A )
78	75, 77	sylan9eq 2525	. . . . . . . . 9 |- ( ( P = { <. 0 , A >. , <. 1 , B >. } /\ A e. V ) -> ( P ` 0 ) = A )
79	40, 78	mpan 684	. . . . . . . 8 |- ( A e. V -> ( P ` 0 ) = A )
80	79	adantr 472	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> ( P ` 0 ) = A )
81	80	3ad2ant2 1052	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( P ` 0 ) = A )
82		fveq1 5878	. . . . . . . . . 10 |- ( P = { <. 0 , A >. , <. 1 , B >. } -> ( P ` 1 ) = ( { <. 0 , A >. , <. 1 , B >. } ` 1 ) )
83		1ex 9656	. . . . . . . . . . 11 |- 1 e. _V
84		fvpr2g 6126	. . . . . . . . . . 11 |- ( ( 1 e. _V /\ B e. V /\ 0 =/= 1 ) -> ( { <. 0 , A >. , <. 1 , B >. } ` 1 ) = B )
85	83, 44, 84	mp3an13 1381	. . . . . . . . . 10 |- ( B e. V -> ( { <. 0 , A >. , <. 1 , B >. } ` 1 ) = B )
86	82, 85	sylan9eq 2525	. . . . . . . . 9 |- ( ( P = { <. 0 , A >. , <. 1 , B >. } /\ B e. V ) -> ( P ` 1 ) = B )
87	40, 86	mpan 684	. . . . . . . 8 |- ( B e. V -> ( P ` 1 ) = B )
88	87	adantl 473	. . . . . . 7 |- ( ( A e. V /\ B e. V ) -> ( P ` 1 ) = B )
89	88	3ad2ant2 1052	. . . . . 6 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( P ` 1 ) = B )
90	81, 89	preq12d 4050	. . . . 5 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
91	68, 74, 90	3eqtr4d 2515	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } )
92		fveq2 5879	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
93	92	fveq2d 5883	. . . . . 6 |- ( k = 0 -> ( E ` ( F ` k ) ) = ( E ` ( F ` 0 ) ) )
94		fveq2 5879	. . . . . . 7 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
95		oveq1 6315	. . . . . . . . 9 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
96	95, 46	syl6eq 2521	. . . . . . . 8 |- ( k = 0 -> ( k + 1 ) = 1 )
97	96	fveq2d 5883	. . . . . . 7 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
98	94, 97	preq12d 4050	. . . . . 6 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
99	93, 98	eqeq12d 2486	. . . . 5 |- ( k = 0 -> ( ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } ) )
100	2, 99	ralsn 4001	. . . 4 |- ( A. k e. { 0 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> ( E ` ( F ` 0 ) ) = { ( P ` 0 ) , ( P ` 1 ) } )
101	91, 100	sylibr 217	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> A. k e. { 0 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
102	14	a1i 11	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( 0..^ ( # ` F ) ) = { 0 } )
103	102	raleqdv 2979	. . 3 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( A. k e. ( 0..^ ( # ` F ) ) ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } <-> A. k e. { 0 } ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) )
104	101, 103	mpbird 240	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> A. k e. ( 0..^ ( # ` F ) ) ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } )
105		snex 4641	. . . . 5 |- { <. 0 , i >. } e. _V
106	1, 105	eqeltri 2545	. . . 4 |- F e. _V
107		prex 4642	. . . . 5 |- { <. 0 , A >. , <. 1 , B >. } e. _V
108	40, 107	eqeltri 2545	. . . 4 |- P e. _V
109		istrl2 25347	. . . 4 |- ( ( ( V e. X /\ E e. Y ) /\ ( F e. _V /\ P e. _V ) ) -> ( F ( V Trails E ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
110	106, 108, 109	mpanr12 699	. . 3 |- ( ( V e. X /\ E e. Y ) -> ( F ( V Trails E ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
111	110	3ad2ant1 1051	. 2 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> ( F ( V Trails E ) P <-> ( F : ( 0..^ ( # ` F ) ) -1-1-> dom E /\ P : ( 0 ... ( # ` F ) ) --> V /\ A. k e. ( 0..^ ( # ` F ) ) ( E ` ( F ` k ) ) = { ( P ` k ) , ( P ` ( k + 1 ) ) } ) ) )
112	39, 66, 104, 111	mpbir3and 1213	1 |- ( ( ( V e. X /\ E e. Y ) /\ ( A e. V /\ B e. V ) /\ ( E ` i ) = { A , B } ) -> F ( V Trails E ) P )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   C_ wss 3390   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   dom cdm 4839   -->wf 5585   -1-1->wf1 5586   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560   ZZcz 10961   ...cfz 11810  ..^cfzo 11942   #chash 12553   Trails ctrail 25306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-wlk 25315  df-trail 25316

This theorem is referenced by:  1pthon  25400