Theorem hashbc 12319

Description: The binomial coefficient counts the number of subsets of a finite set of a given size. This is Metamath 100 proof #58 (formula for the number of combinations). (Contributed by Mario Carneiro, 13-Jul-2014.)

Assertion

Ref	Expression
hashbc	|- ( ( A e. Fin /\ K e. ZZ ) -> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )

Distinct variable groups:   x, A   x, K

Proof of Theorem hashbc

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

Step	Hyp	Ref	Expression
1		fveq2 5794	. . . . . 6 |- ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
2	1	oveq1d 6210	. . . . 5 |- ( w = (/) -> ( ( # ` w ) _C k ) = ( ( # ` (/) ) _C k ) )
3		pweq 3966	. . . . . . 7 |- ( w = (/) -> ~P w = ~P (/) )
4		rabeq 3066	. . . . . . 7 |- ( ~P w = ~P (/) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P (/) | ( # ` x ) = k } )
5	3, 4	syl 16	. . . . . 6 |- ( w = (/) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P (/) | ( # ` x ) = k } )
6	5	fveq2d 5798	. . . . 5 |- ( w = (/) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
7	2, 6	eqeq12d 2474	. . . 4 |- ( w = (/) -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) ) )
8	7	ralbidv 2843	. . 3 |- ( w = (/) -> ( A. k e. ZZ ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> A. k e. ZZ ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) ) )
9		fveq2 5794	. . . . . 6 |- ( w = y -> ( # ` w ) = ( # ` y ) )
10	9	oveq1d 6210	. . . . 5 |- ( w = y -> ( ( # ` w ) _C k ) = ( ( # ` y ) _C k ) )
11		pweq 3966	. . . . . . 7 |- ( w = y -> ~P w = ~P y )
12		rabeq 3066	. . . . . . 7 |- ( ~P w = ~P y -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = k } )
13	11, 12	syl 16	. . . . . 6 |- ( w = y -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = k } )
14	13	fveq2d 5798	. . . . 5 |- ( w = y -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) )
15	10, 14	eqeq12d 2474	. . . 4 |- ( w = y -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) ) )
16	15	ralbidv 2843	. . 3 |- ( w = y -> ( A. k e. ZZ ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> A. k e. ZZ ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) ) )
17		fveq2 5794	. . . . . 6 |- ( w = ( y u. { z } ) -> ( # ` w ) = ( # ` ( y u. { z } ) ) )
18	17	oveq1d 6210	. . . . 5 |- ( w = ( y u. { z } ) -> ( ( # ` w ) _C k ) = ( ( # ` ( y u. { z } ) ) _C k ) )
19		pweq 3966	. . . . . . 7 |- ( w = ( y u. { z } ) -> ~P w = ~P ( y u. { z } ) )
20		rabeq 3066	. . . . . . 7 |- ( ~P w = ~P ( y u. { z } ) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P ( y u. { z } ) | ( # ` x ) = k } )
21	19, 20	syl 16	. . . . . 6 |- ( w = ( y u. { z } ) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P ( y u. { z } ) | ( # ` x ) = k } )
22	21	fveq2d 5798	. . . . 5 |- ( w = ( y u. { z } ) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) )
23	18, 22	eqeq12d 2474	. . . 4 |- ( w = ( y u. { z } ) -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) ) )
24	23	ralbidv 2843	. . 3 |- ( w = ( y u. { z } ) -> ( A. k e. ZZ ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> A. k e. ZZ ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) ) )
25		fveq2 5794	. . . . . 6 |- ( w = A -> ( # ` w ) = ( # ` A ) )
26	25	oveq1d 6210	. . . . 5 |- ( w = A -> ( ( # ` w ) _C k ) = ( ( # ` A ) _C k ) )
27		pweq 3966	. . . . . . 7 |- ( w = A -> ~P w = ~P A )
28		rabeq 3066	. . . . . . 7 |- ( ~P w = ~P A -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = k } )
29	27, 28	syl 16	. . . . . 6 |- ( w = A -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = k } )
30	29	fveq2d 5798	. . . . 5 |- ( w = A -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
31	26, 30	eqeq12d 2474	. . . 4 |- ( w = A -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) ) )
32	31	ralbidv 2843	. . 3 |- ( w = A -> ( A. k e. ZZ ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> A. k e. ZZ ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) ) )
33		hash0 12247	. . . . . . . . . 10 |- ( # ` (/) ) = 0
34	33	a1i 11	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> ( # ` (/) ) = 0 )
35		elfz1eq 11574	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> k = 0 )
36	34, 35	oveq12d 6213	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( 0 _C 0 ) )
37		0nn0 10700	. . . . . . . . 9 |- 0 e. NN0
38		bcn0 12198	. . . . . . . . 9 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
39	37, 38	ax-mp 5	. . . . . . . 8 |- ( 0 _C 0 ) = 1
40	36, 39	syl6eq 2509	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = 1 )
41		pw0 4123	. . . . . . . . . 10 |- ~P (/) = { (/) }
42	35	eqcomd 2460	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 0 ) -> 0 = k )
43	41	raleqi 3021	. . . . . . . . . . . . 13 |- ( A. x e. ~P (/) ( # ` x ) = k <-> A. x e. { (/) } ( # ` x ) = k )
44		0ex 4525	. . . . . . . . . . . . . 14 |- (/) e. _V
45		fveq2 5794	. . . . . . . . . . . . . . . 16 |- ( x = (/) -> ( # ` x ) = ( # ` (/) ) )
46	45, 33	syl6eq 2509	. . . . . . . . . . . . . . 15 |- ( x = (/) -> ( # ` x ) = 0 )
47	46	eqeq1d 2454	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( ( # ` x ) = k <-> 0 = k ) )
48	44, 47	ralsn 4018	. . . . . . . . . . . . 13 |- ( A. x e. { (/) } ( # ` x ) = k <-> 0 = k )
49	43, 48	bitri 249	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) ( # ` x ) = k <-> 0 = k )
50	42, 49	sylibr 212	. . . . . . . . . . 11 |- ( k e. ( 0 ... 0 ) -> A. x e. ~P (/) ( # ` x ) = k )
51		rabid2 2998	. . . . . . . . . . 11 |- ( ~P (/) = { x e. ~P (/) | ( # ` x ) = k } <-> A. x e. ~P (/) ( # ` x ) = k )
52	50, 51	sylibr 212	. . . . . . . . . 10 |- ( k e. ( 0 ... 0 ) -> ~P (/) = { x e. ~P (/) | ( # ` x ) = k } )
53	41, 52	syl5reqr 2508	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> { x e. ~P (/) | ( # ` x ) = k } = { (/) } )
54	53	fveq2d 5798	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` { (/) } ) )
55		hashsng 12248	. . . . . . . . 9 |- ( (/) e. _V -> ( # ` { (/) } ) = 1 )
56	44, 55	ax-mp 5	. . . . . . . 8 |- ( # ` { (/) } ) = 1
57	54, 56	syl6eq 2509	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 1 )
58	40, 57	eqtr4d 2496	. . . . . 6 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
59	58	adantl 466	. . . . 5 |- ( ( k e. ZZ /\ k e. ( 0 ... 0 ) ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
60	33	oveq1i 6205	. . . . . 6 |- ( ( # ` (/) ) _C k ) = ( 0 _C k )
61		bcval3 12194	. . . . . . . 8 |- ( ( 0 e. NN0 /\ k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
62	37, 61	mp3an1 1302	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
63		id 22	. . . . . . . . . . . . . 14 |- ( 0 = k -> 0 = k )
64		0z 10763	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
65		elfz3 11573	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> 0 e. ( 0 ... 0 ) )
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 |- 0 e. ( 0 ... 0 )
67	63, 66	syl6eqelr 2549	. . . . . . . . . . . . 13 |- ( 0 = k -> k e. ( 0 ... 0 ) )
68	67	con3i 135	. . . . . . . . . . . 12 |- ( -. k e. ( 0 ... 0 ) -> -. 0 = k )
69	68	adantl 466	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> -. 0 = k )
70	41	raleqi 3021	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) -. ( # ` x ) = k <-> A. x e. { (/) } -. ( # ` x ) = k )
71	47	notbid 294	. . . . . . . . . . . . 13 |- ( x = (/) -> ( -. ( # ` x ) = k <-> -. 0 = k ) )
72	44, 71	ralsn 4018	. . . . . . . . . . . 12 |- ( A. x e. { (/) } -. ( # ` x ) = k <-> -. 0 = k )
73	70, 72	bitri 249	. . . . . . . . . . 11 |- ( A. x e. ~P (/) -. ( # ` x ) = k <-> -. 0 = k )
74	69, 73	sylibr 212	. . . . . . . . . 10 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> A. x e. ~P (/) -. ( # ` x ) = k )
75		rabeq0 3762	. . . . . . . . . 10 |- ( { x e. ~P (/) | ( # ` x ) = k } = (/) <-> A. x e. ~P (/) -. ( # ` x ) = k )
76	74, 75	sylibr 212	. . . . . . . . 9 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> { x e. ~P (/) | ( # ` x ) = k } = (/) )
77	76	fveq2d 5798	. . . . . . . 8 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` (/) ) )
78	77, 33	syl6eq 2509	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 0 )
79	62, 78	eqtr4d 2496	. . . . . 6 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
80	60, 79	syl5eq 2505	. . . . 5 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
81	59, 80	pm2.61dan 789	. . . 4 |- ( k e. ZZ -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
82	81	rgen 2893	. . 3 |- A. k e. ZZ ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } )
83		oveq2 6203	. . . . . 6 |- ( k = j -> ( ( # ` y ) _C k ) = ( ( # ` y ) _C j ) )
84		eqeq2 2467	. . . . . . . . 9 |- ( k = j -> ( ( # ` x ) = k <-> ( # ` x ) = j ) )
85	84	rabbidv 3064	. . . . . . . 8 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = j } )
86		fveq2 5794	. . . . . . . . . 10 |- ( x = z -> ( # ` x ) = ( # ` z ) )
87	86	eqeq1d 2454	. . . . . . . . 9 |- ( x = z -> ( ( # ` x ) = j <-> ( # ` z ) = j ) )
88	87	cbvrabv 3071	. . . . . . . 8 |- { x e. ~P y | ( # ` x ) = j } = { z e. ~P y | ( # ` z ) = j }
89	85, 88	syl6eq 2509	. . . . . . 7 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { z e. ~P y | ( # ` z ) = j } )
90	89	fveq2d 5798	. . . . . 6 |- ( k = j -> ( # ` { x e. ~P y | ( # ` x ) = k } ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
91	83, 90	eqeq12d 2474	. . . . 5 |- ( k = j -> ( ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) )
92	91	cbvralv 3047	. . . 4 |- ( A. k e. ZZ ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) <-> A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
93		simpll 753	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> y e. Fin )
94		simplr 754	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> -. z e. y )
95		simprr 756	. . . . . . . 8 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
96	88	fveq2i 5797	. . . . . . . . . 10 |- ( # ` { x e. ~P y | ( # ` x ) = j } ) = ( # ` { z e. ~P y | ( # ` z ) = j } )
97	96	eqeq2i 2470	. . . . . . . . 9 |- ( ( ( # ` y ) _C j ) = ( # ` { x e. ~P y | ( # ` x ) = j } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
98	97	ralbii 2836	. . . . . . . 8 |- ( A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { x e. ~P y | ( # ` x ) = j } ) <-> A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
99	95, 98	sylibr 212	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { x e. ~P y | ( # ` x ) = j } ) )
100		simprl 755	. . . . . . 7 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> k e. ZZ )
101	93, 94, 99, 100	hashbclem 12318	. . . . . 6 |- ( ( ( y e. Fin /\ -. z e. y ) /\ ( k e. ZZ /\ A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) ) -> ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) )
102	101	expr 615	. . . . 5 |- ( ( ( y e. Fin /\ -. z e. y ) /\ k e. ZZ ) -> ( A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) -> ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) ) )
103	102	ralrimdva 2906	. . . 4 |- ( ( y e. Fin /\ -. z e. y ) -> ( A. j e. ZZ ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) -> A. k e. ZZ ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) ) )
104	92, 103	syl5bi 217	. . 3 |- ( ( y e. Fin /\ -. z e. y ) -> ( A. k e. ZZ ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) -> A. k e. ZZ ( ( # ` ( y u. { z } ) ) _C k ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) ) )
105	8, 16, 24, 32, 82, 104	findcard2s 7659	. 2 |- ( A e. Fin -> A. k e. ZZ ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
106		oveq2 6203	. . . 4 |- ( k = K -> ( ( # ` A ) _C k ) = ( ( # ` A ) _C K ) )
107		eqeq2 2467	. . . . . 6 |- ( k = K -> ( ( # ` x ) = k <-> ( # ` x ) = K ) )
108	107	rabbidv 3064	. . . . 5 |- ( k = K -> { x e. ~P A | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = K } )
109	108	fveq2d 5798	. . . 4 |- ( k = K -> ( # ` { x e. ~P A | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )
110	106, 109	eqeq12d 2474	. . 3 |- ( k = K -> ( ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) <-> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) ) )
111	110	rspccva 3172	. 2 |- ( ( A. k e. ZZ ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) /\ K e. ZZ ) -> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )
112	105, 111	sylan 471	1 |- ( ( A e. Fin /\ K e. ZZ ) -> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   A.wral 2796   {crab 2800   _Vcvv 3072   u. cun 3429   (/)c0 3740   ~Pcpw 3963   {csn 3980   ` cfv 5521  (class class class)co 6195   Fincfn 7415   0cc0 9388   1c1 9389   NN0cn0 10685   ZZcz 10752   ...cfz 11549   _C cbc 12190   #chash 12215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-1o 7025  df-2o 7026  df-oadd 7029  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-fin 7419  df-card 8215  df-cda 8443  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-div 10100  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-rp 11098  df-fz 11550  df-seq 11919  df-fac 12164  df-bc 12191  df-hash 12216

This theorem is referenced by:  hashbc2  14180  sylow1lem1  16213  musum  22659  ballotlem1  27008  ballotlem2  27010