Theorem hashbc 12623

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 5870	. . . . . 6 |- ( w = (/) -> ( # ` w ) = ( # ` (/) ) )
2	1	oveq1d 6310	. . . . 5 |- ( w = (/) -> ( ( # ` w ) _C k ) = ( ( # ` (/) ) _C k ) )
3		pweq 3956	. . . . . . 7 |- ( w = (/) -> ~P w = ~P (/) )
4		rabeq 3040	. . . . . . 7 |- ( ~P w = ~P (/) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P (/) | ( # ` x ) = k } )
5	3, 4	syl 17	. . . . . 6 |- ( w = (/) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P (/) | ( # ` x ) = k } )
6	5	fveq2d 5874	. . . . 5 |- ( w = (/) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
7	2, 6	eqeq12d 2468	. . . 4 |- ( w = (/) -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) ) )
8	7	ralbidv 2829	. . 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 5870	. . . . . 6 |- ( w = y -> ( # ` w ) = ( # ` y ) )
10	9	oveq1d 6310	. . . . 5 |- ( w = y -> ( ( # ` w ) _C k ) = ( ( # ` y ) _C k ) )
11		pweq 3956	. . . . . . 7 |- ( w = y -> ~P w = ~P y )
12		rabeq 3040	. . . . . . 7 |- ( ~P w = ~P y -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = k } )
13	11, 12	syl 17	. . . . . 6 |- ( w = y -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = k } )
14	13	fveq2d 5874	. . . . 5 |- ( w = y -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) )
15	10, 14	eqeq12d 2468	. . . 4 |- ( w = y -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) ) )
16	15	ralbidv 2829	. . 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 5870	. . . . . 6 |- ( w = ( y u. { z } ) -> ( # ` w ) = ( # ` ( y u. { z } ) ) )
18	17	oveq1d 6310	. . . . 5 |- ( w = ( y u. { z } ) -> ( ( # ` w ) _C k ) = ( ( # ` ( y u. { z } ) ) _C k ) )
19		pweq 3956	. . . . . . 7 |- ( w = ( y u. { z } ) -> ~P w = ~P ( y u. { z } ) )
20		rabeq 3040	. . . . . . 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 17	. . . . . 6 |- ( w = ( y u. { z } ) -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P ( y u. { z } ) | ( # ` x ) = k } )
22	21	fveq2d 5874	. . . . 5 |- ( w = ( y u. { z } ) -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P ( y u. { z } ) | ( # ` x ) = k } ) )
23	18, 22	eqeq12d 2468	. . . 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 2829	. . 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 5870	. . . . . 6 |- ( w = A -> ( # ` w ) = ( # ` A ) )
26	25	oveq1d 6310	. . . . 5 |- ( w = A -> ( ( # ` w ) _C k ) = ( ( # ` A ) _C k ) )
27		pweq 3956	. . . . . . 7 |- ( w = A -> ~P w = ~P A )
28		rabeq 3040	. . . . . . 7 |- ( ~P w = ~P A -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = k } )
29	27, 28	syl 17	. . . . . 6 |- ( w = A -> { x e. ~P w | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = k } )
30	29	fveq2d 5874	. . . . 5 |- ( w = A -> ( # ` { x e. ~P w | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
31	26, 30	eqeq12d 2468	. . . 4 |- ( w = A -> ( ( ( # ` w ) _C k ) = ( # ` { x e. ~P w | ( # ` x ) = k } ) <-> ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) ) )
32	31	ralbidv 2829	. . 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 12555	. . . . . . . . . 10 |- ( # ` (/) ) = 0
34	33	a1i 11	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> ( # ` (/) ) = 0 )
35		elfz1eq 11817	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> k = 0 )
36	34, 35	oveq12d 6313	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( 0 _C 0 ) )
37		0nn0 10891	. . . . . . . . 9 |- 0 e. NN0
38		bcn0 12502	. . . . . . . . 9 |- ( 0 e. NN0 -> ( 0 _C 0 ) = 1 )
39	37, 38	ax-mp 5	. . . . . . . 8 |- ( 0 _C 0 ) = 1
40	36, 39	syl6eq 2503	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = 1 )
41		pw0 4122	. . . . . . . . . 10 |- ~P (/) = { (/) }
42	35	eqcomd 2459	. . . . . . . . . . . 12 |- ( k e. ( 0 ... 0 ) -> 0 = k )
43	41	raleqi 2993	. . . . . . . . . . . . 13 |- ( A. x e. ~P (/) ( # ` x ) = k <-> A. x e. { (/) } ( # ` x ) = k )
44		0ex 4538	. . . . . . . . . . . . . 14 |- (/) e. _V
45		fveq2 5870	. . . . . . . . . . . . . . . 16 |- ( x = (/) -> ( # ` x ) = ( # ` (/) ) )
46	45, 33	syl6eq 2503	. . . . . . . . . . . . . . 15 |- ( x = (/) -> ( # ` x ) = 0 )
47	46	eqeq1d 2455	. . . . . . . . . . . . . 14 |- ( x = (/) -> ( ( # ` x ) = k <-> 0 = k ) )
48	44, 47	ralsn 4012	. . . . . . . . . . . . 13 |- ( A. x e. { (/) } ( # ` x ) = k <-> 0 = k )
49	43, 48	bitri 253	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) ( # ` x ) = k <-> 0 = k )
50	42, 49	sylibr 216	. . . . . . . . . . 11 |- ( k e. ( 0 ... 0 ) -> A. x e. ~P (/) ( # ` x ) = k )
51		rabid2 2970	. . . . . . . . . . 11 |- ( ~P (/) = { x e. ~P (/) | ( # ` x ) = k } <-> A. x e. ~P (/) ( # ` x ) = k )
52	50, 51	sylibr 216	. . . . . . . . . 10 |- ( k e. ( 0 ... 0 ) -> ~P (/) = { x e. ~P (/) | ( # ` x ) = k } )
53	41, 52	syl5reqr 2502	. . . . . . . . 9 |- ( k e. ( 0 ... 0 ) -> { x e. ~P (/) | ( # ` x ) = k } = { (/) } )
54	53	fveq2d 5874	. . . . . . . 8 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` { (/) } ) )
55		hashsng 12556	. . . . . . . . 9 |- ( (/) e. _V -> ( # ` { (/) } ) = 1 )
56	44, 55	ax-mp 5	. . . . . . . 8 |- ( # ` { (/) } ) = 1
57	54, 56	syl6eq 2503	. . . . . . 7 |- ( k e. ( 0 ... 0 ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 1 )
58	40, 57	eqtr4d 2490	. . . . . 6 |- ( k e. ( 0 ... 0 ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
59	58	adantl 468	. . . . 5 |- ( ( k e. ZZ /\ k e. ( 0 ... 0 ) ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
60	33	oveq1i 6305	. . . . . 6 |- ( ( # ` (/) ) _C k ) = ( 0 _C k )
61		bcval3 12498	. . . . . . . 8 |- ( ( 0 e. NN0 /\ k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
62	37, 61	mp3an1 1353	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = 0 )
63		id 22	. . . . . . . . . . . . . 14 |- ( 0 = k -> 0 = k )
64		0z 10955	. . . . . . . . . . . . . . 15 |- 0 e. ZZ
65		elfz3 11816	. . . . . . . . . . . . . . 15 |- ( 0 e. ZZ -> 0 e. ( 0 ... 0 ) )
66	64, 65	ax-mp 5	. . . . . . . . . . . . . 14 |- 0 e. ( 0 ... 0 )
67	63, 66	syl6eqelr 2540	. . . . . . . . . . . . 13 |- ( 0 = k -> k e. ( 0 ... 0 ) )
68	67	con3i 141	. . . . . . . . . . . 12 |- ( -. k e. ( 0 ... 0 ) -> -. 0 = k )
69	68	adantl 468	. . . . . . . . . . 11 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> -. 0 = k )
70	41	raleqi 2993	. . . . . . . . . . . 12 |- ( A. x e. ~P (/) -. ( # ` x ) = k <-> A. x e. { (/) } -. ( # ` x ) = k )
71	47	notbid 296	. . . . . . . . . . . . 13 |- ( x = (/) -> ( -. ( # ` x ) = k <-> -. 0 = k ) )
72	44, 71	ralsn 4012	. . . . . . . . . . . 12 |- ( A. x e. { (/) } -. ( # ` x ) = k <-> -. 0 = k )
73	70, 72	bitri 253	. . . . . . . . . . 11 |- ( A. x e. ~P (/) -. ( # ` x ) = k <-> -. 0 = k )
74	69, 73	sylibr 216	. . . . . . . . . 10 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> A. x e. ~P (/) -. ( # ` x ) = k )
75		rabeq0 3756	. . . . . . . . . 10 |- ( { x e. ~P (/) | ( # ` x ) = k } = (/) <-> A. x e. ~P (/) -. ( # ` x ) = k )
76	74, 75	sylibr 216	. . . . . . . . 9 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> { x e. ~P (/) | ( # ` x ) = k } = (/) )
77	76	fveq2d 5874	. . . . . . . 8 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = ( # ` (/) ) )
78	77, 33	syl6eq 2503	. . . . . . 7 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( # ` { x e. ~P (/) | ( # ` x ) = k } ) = 0 )
79	62, 78	eqtr4d 2490	. . . . . 6 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( 0 _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
80	60, 79	syl5eq 2499	. . . . 5 |- ( ( k e. ZZ /\ -. k e. ( 0 ... 0 ) ) -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
81	59, 80	pm2.61dan 801	. . . 4 |- ( k e. ZZ -> ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } ) )
82	81	rgen 2749	. . 3 |- A. k e. ZZ ( ( # ` (/) ) _C k ) = ( # ` { x e. ~P (/) | ( # ` x ) = k } )
83		oveq2 6303	. . . . . 6 |- ( k = j -> ( ( # ` y ) _C k ) = ( ( # ` y ) _C j ) )
84		eqeq2 2464	. . . . . . . . 9 |- ( k = j -> ( ( # ` x ) = k <-> ( # ` x ) = j ) )
85	84	rabbidv 3038	. . . . . . . 8 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { x e. ~P y | ( # ` x ) = j } )
86		fveq2 5870	. . . . . . . . . 10 |- ( x = z -> ( # ` x ) = ( # ` z ) )
87	86	eqeq1d 2455	. . . . . . . . 9 |- ( x = z -> ( ( # ` x ) = j <-> ( # ` z ) = j ) )
88	87	cbvrabv 3046	. . . . . . . 8 |- { x e. ~P y | ( # ` x ) = j } = { z e. ~P y | ( # ` z ) = j }
89	85, 88	syl6eq 2503	. . . . . . 7 |- ( k = j -> { x e. ~P y | ( # ` x ) = k } = { z e. ~P y | ( # ` z ) = j } )
90	89	fveq2d 5874	. . . . . 6 |- ( k = j -> ( # ` { x e. ~P y | ( # ` x ) = k } ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
91	83, 90	eqeq12d 2468	. . . . 5 |- ( k = j -> ( ( ( # ` y ) _C k ) = ( # ` { x e. ~P y | ( # ` x ) = k } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) ) )
92	91	cbvralv 3021	. . . 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 761	. . . . . . 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 763	. . . . . . 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 767	. . . . . . . 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 5873	. . . . . . . . . 10 |- ( # ` { x e. ~P y | ( # ` x ) = j } ) = ( # ` { z e. ~P y | ( # ` z ) = j } )
97	96	eqeq2i 2465	. . . . . . . . 9 |- ( ( ( # ` y ) _C j ) = ( # ` { x e. ~P y | ( # ` x ) = j } ) <-> ( ( # ` y ) _C j ) = ( # ` { z e. ~P y | ( # ` z ) = j } ) )
98	97	ralbii 2821	. . . . . . . 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 216	. . . . . . 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 765	. . . . . . 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 12622	. . . . . 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 620	. . . . 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 2808	. . . 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 221	. . 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 7817	. 2 |- ( A e. Fin -> A. k e. ZZ ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) )
106		oveq2 6303	. . . 4 |- ( k = K -> ( ( # ` A ) _C k ) = ( ( # ` A ) _C K ) )
107		eqeq2 2464	. . . . . 6 |- ( k = K -> ( ( # ` x ) = k <-> ( # ` x ) = K ) )
108	107	rabbidv 3038	. . . . 5 |- ( k = K -> { x e. ~P A | ( # ` x ) = k } = { x e. ~P A | ( # ` x ) = K } )
109	108	fveq2d 5874	. . . 4 |- ( k = K -> ( # ` { x e. ~P A | ( # ` x ) = k } ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )
110	106, 109	eqeq12d 2468	. . 3 |- ( k = K -> ( ( ( # ` A ) _C k ) = ( # ` { x e. ~P A | ( # ` x ) = k } ) <-> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) ) )
111	110	rspccva 3151	. 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 474	1 |- ( ( A e. Fin /\ K e. ZZ ) -> ( ( # ` A ) _C K ) = ( # ` { x e. ~P A | ( # ` x ) = K } ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   {crab 2743   _Vcvv 3047   u. cun 3404   (/)c0 3733   ~Pcpw 3953   {csn 3970   ` cfv 5585  (class class class)co 6295   Fincfn 7574   0cc0 9544   1c1 9545   NN0cn0 10876   ZZcz 10944   ...cfz 11791   _C cbc 12494   #chash 12522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-2o 7188  df-oadd 7191  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-cda 8603  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-div 10277  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-rp 11310  df-fz 11792  df-seq 12221  df-fac 12467  df-bc 12495  df-hash 12523

This theorem is referenced by:  hashbc2  14970  sylow1lem1  17262  musum  24132  ballotlem1  29331  ballotlem2  29333