Theorem hashbc 12406

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 5774	. . . . . 6 |- ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
2	1	oveq1d 6211	. . . . 5 |- ( w = (/) -> ( ( # ` w ) _C k ) = ( ( # ` (/) ) _C k ) )
3		pweq 3930	. . . . . . 7 |- ( w = (/) -> ~P w = ~P (/) )
4		rabeq 3028	. . . . . . 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 5778	. . . . 5 |- ( w = (/) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
7	2, 6	eqeq12d 2404	. . . 4 |- ( w = (/) -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) ) )
8	7	ralbidv 2821	. . 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 5774	. . . . . 6 |- ( w = y -> ( # ` w ) = ( # ` y ) )
10	9	oveq1d 6211	. . . . 5 |- ( w = y -> ( ( # ` w ) _C k ) = ( ( # ` y ) _C k ) )
11		pweq 3930	. . . . . . 7 |- ( w = y -> ~P w = ~P y )
12		rabeq 3028	. . . . . . 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 5778	. . . . 5 |- ( w = y -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) )
15	10, 14	eqeq12d 2404	. . . 4 |- ( w = y -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) ) )
16	15	ralbidv 2821	. . 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 5774	. . . . . 6 |- ( w = ( y u. { z } ) -> ( # ` w ) = ( # ` ( y u. { z } ) ) )
18	17	oveq1d 6211	. . . . 5 |- ( w = ( y u. { z } ) -> ( ( # ` w ) _C k ) = ( ( # ` ( y u. { z } ) ) _C k ) )
19		pweq 3930	. . . . . . 7 |- ( w = ( y u. { z } ) -> ~P w = ~P ( y u. { z } ) )
20		rabeq 3028	. . . . . . 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 5778	. . . . 5 |- ( w = ( y u. { z } ) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) )
23	18, 22	eqeq12d 2404	. . . 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 2821	. . 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 5774	. . . . . 6 |- ( w = A -> ( # ` w ) = ( # ` A ) )
26	25	oveq1d 6211	. . . . 5 |- ( w = A -> ( ( # ` w ) _C k ) = ( ( # ` A ) _C k ) )
27		pweq 3930	. . . . . . 7 |- ( w = A -> ~P w = ~P A )
28		rabeq 3028	. . . . . . 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 5778	. . . . 5 |- ( w = A -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
31	26, 30	eqeq12d 2404	. . . 4 |- ( w = A -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) ) )
32	31	ralbidv 2821	. . 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 12340	. . . . . . . . . 10 |- ( # ` (/) ) = 0
34	33	a1i 11	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> ( # ` (/) ) = 0 )
35		elfz1eq 11618	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> k = 0 )
36	34, 35	oveq12d 6214	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( 0 _C 0 ) )
37		0nn0 10727	. . . . . . . . 9 |- 0 e. NN0
38		bcn0 12290	. . . . . . . . 9 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
39	37, 38	ax-mp 5	. . . . . . . 8 |- ( 0 _C 0 ) = 1
40	36, 39	syl6eq 2439	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = 1 )
41		pw0 4091	. . . . . . . . . 10 |- ~P (/) = { (/) }
42	35	eqcomd 2390	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 0 ) -> 0 = k )
43	41	raleqi 2983	. . . . . . . . . . . . 13 |- ( A. x e. ~P (/) ( # ` x ) = k <-> A. x e. { (/) } ( # ` x ) = k )
44		0ex 4497	. . . . . . . . . . . . . 14 |- (/) e. _V
45		fveq2 5774	. . . . . . . . . . . . . . . 16 |- ( x = (/) -> ( # ` x ) = ( # ` (/) ) )
46	45, 33	syl6eq 2439	. . . . . . . . . . . . . . 15 |- ( x = (/) -> ( # ` x ) = 0 )
47	46	eqeq1d 2384	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( ( # ` x ) = k <-> 0 = k ) )
48	44, 47	ralsn 3983	. . . . . . . . . . . . 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 2960	. . . . . . . . . . 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 2438	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> { x e. ~P (/) | ( # ` x ) = k } = { (/) } )
54	53	fveq2d 5778	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` { (/) } ) )
55		hashsng 12341	. . . . . . . . 9 |- ( (/) e. _V -> ( # ` { (/) } ) = 1 )
56	44, 55	ax-mp 5	. . . . . . . 8 |- ( # ` { (/) } ) = 1
57	54, 56	syl6eq 2439	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 1 )
58	40, 57	eqtr4d 2426	. . . . . 6 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
59	58	adantl 464	. . . . 5 |- ( ( k e. ZZ /\ k e. ( 0 ... 0 ) ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
60	33	oveq1i 6206	. . . . . 6 |- ( ( # ` (/) ) _C k ) = ( 0 _C k )
61		bcval3 12286	. . . . . . . 8 |- ( ( 0 e. NN0 /\ k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
62	37, 61	mp3an1 1309	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
63		id 22	. . . . . . . . . . . . . 14 |- ( 0 = k -> 0 = k )
64		0z 10792	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
65		elfz3 11617	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> 0 e. ( 0 ... 0 ) )
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 |- 0 e. ( 0 ... 0 )
67	63, 66	syl6eqelr 2479	. . . . . . . . . . . . 13 |- ( 0 = k -> k e. ( 0 ... 0 ) )
68	67	con3i 135	. . . . . . . . . . . 12 |- ( -. k e. ( 0 ... 0 ) -> -. 0 = k )
69	68	adantl 464	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> -. 0 = k )
70	41	raleqi 2983	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) -. ( # ` x ) = k <-> A. x e. { (/) } -. ( # ` x ) = k )
71	47	notbid 292	. . . . . . . . . . . . 13 |- ( x = (/) -> ( -. ( # ` x ) = k <-> -. 0 = k ) )
72	44, 71	ralsn 3983	. . . . . . . . . . . 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 3734	. . . . . . . . . 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 5778	. . . . . . . 8 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` (/) ) )
78	77, 33	syl6eq 2439	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 0 )
79	62, 78	eqtr4d 2426	. . . . . 6 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
80	60, 79	syl5eq 2435	. . . . 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 2742	. . 3 |- A. k e. ZZ ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } )
83		oveq2 6204	. . . . . 6 |- ( k = j -> ( ( # ` y ) _C k ) = ( ( # ` y ) _C j ) )
84		eqeq2 2397	. . . . . . . . 9 |- ( k = j -> ( ( # ` x ) = k <-> ( # ` x ) = j ) )
85	84	rabbidv 3026	. . . . . . . 8 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = j } )
86		fveq2 5774	. . . . . . . . . 10 |- ( x = z -> ( # ` x ) = ( # ` z ) )
87	86	eqeq1d 2384	. . . . . . . . 9 |- ( x = z -> ( ( # ` x ) = j <-> ( # ` z ) = j ) )
88	87	cbvrabv 3033	. . . . . . . 8 |- { x e. ~P y | ( # ` x ) = j } = { z e. ~P y | ( # ` z ) = j }
89	85, 88	syl6eq 2439	. . . . . . 7 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { z e. ~P y | ( # ` z ) = j } )
90	89	fveq2d 5778	. . . . . 6 |- ( k = j -> ( # ` { x e. ~P y | ( # ` x ) = k } ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
91	83, 90	eqeq12d 2404	. . . . 5 |- ( k = j -> ( ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) )
92	91	cbvralv 3009	. . . 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 751	. . . . . . 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 753	. . . . . . 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 755	. . . . . . . 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 5777	. . . . . . . . . 10 |- ( # ` { x e. ~P y | ( # ` x ) = j } ) = ( # ` { z e. ~P y | ( # ` z ) = j } )
97	96	eqeq2i 2400	. . . . . . . . 9 |- ( ( ( # ` y ) _C j ) = ( # ` { x e. ~P y | ( # ` x ) = j } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
98	97	ralbii 2813	. . . . . . . 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 754	. . . . . . 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 12405	. . . . . 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 613	. . . . 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 2800	. . . 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 7676	. 2 |- ( A e. Fin -> A. k e. ZZ ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
106		oveq2 6204	. . . 4 |- ( k = K -> ( ( # ` A ) _C k ) = ( ( # ` A ) _C K ) )
107		eqeq2 2397	. . . . . 6 |- ( k = K -> ( ( # ` x ) = k <-> ( # ` x ) = K ) )
108	107	rabbidv 3026	. . . . 5 |- ( k = K -> { x e. ~P A | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = K } )
109	108	fveq2d 5778	. . . 4 |- ( k = K -> ( # ` { x e. ~P A | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )
110	106, 109	eqeq12d 2404	. . 3 |- ( k = K -> ( ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) <-> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) ) )
111	110	rspccva 3134	. 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 469	1 |- ( ( A e. Fin /\ K e. ZZ ) -> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   = wceq 1399   e. wcel 1826   A.wral 2732   {crab 2736   _Vcvv 3034   u. cun 3387   (/)c0 3711   ~Pcpw 3927   {csn 3944   ` cfv 5496  (class class class)co 6196   Fincfn 7435   0cc0 9403   1c1 9404   NN0cn0 10712   ZZcz 10781   ...cfz 11593   _C cbc 12282   #chash 12307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-1o 7048  df-2o 7049  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-card 8233  df-cda 8461  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-rp 11140  df-fz 11594  df-seq 12011  df-fac 12256  df-bc 12283  df-hash 12308

This theorem is referenced by:  hashbc2  14526  sylow1lem1  16735  musum  23584  ballotlem1  28608  ballotlem2  28610